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# Milk Removal: The Key to Maintaining Milk Production and a Tool to Enhance Efficiency

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Submitted: March 14th, 2011 Reviewed: June 18th, 2012 Published: September 26th, 2012

DOI: 10.5772/50773

From the Edited Volume

## Milk Production

Edited by Narongsak Chaiyabutr

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## 1. Introduction

Manipulation of the lactation curve by increasing peak milk production and (or) lactation persistency has consistently been a focus of both researchers and dairy producers. One of the key findings that has enabled manipulation of lactation performance is that the milk yield of dairy cows is responsive to demands of offspring or milk removal; hence milk production can be increased by frequent milking or suckling. Early studies illustrated the galactopoietic effect of frequent milking during the entire lactation, with three times daily milking increasing milk yield by up to 20% relative to twice daily milking. Research using nurse cows revealed a long-term increase in milk production when cows and heifers were allowed to suckle a calf during the first 8 to 10 wk of lactation (Everitt & Phillips, 1971; Edmunds, 1977; Fulkerson, 1981). The results of these experiments laid the groundwork for current research, which has identified a time during early lactation wherein the mammary gland of dairy cows is especially receptive to the stimulus of frequent milking. More recently, it has been established that frequent milk removal (three or more times daily) for a short duration within the first three weeks of lactation can increase milk production through the remainder of lactation (Hale et al., 2003; Dahl et al., 2004b; Wall & McFadden, 2007b). Since the establishment of the galactopoietic effect on milk production, several experiments have been conducted to identify the factors that regulate the milk yield response. These reports have documented consistent responses to increased milking frequency; however, questions remain about the mechanisms involved in regulation of milk production efficiency.

### 1.1. Frequent milking or suckling increases milk production

As previously indicated, frequent milking of dairy cows has emerged as an effective management tool for dairy farmers to increase milk production efficiency. Although it is a relatively novel management practice, the original interest and research in this area dates back to the late 1800’s (Hills, 1890; 1898). Despite considerable variation in the magnitude of the milk yield response, it was recognized long ago that thrice-daily milking (3X) increased milk production relative to twice-daily milking (2X), and that frequent milking could be a profitable management tool if costs associated with the extra milkings are outweighed by the value of additional milk obtained (Riford, 1922; Dahlberg, 1924). Cows milked 3X generally produced about 20% more milk than those milked 2X, and milk production could be increased another 7% by milking four times daily (4X) instead of 3X (Woodward, 1931). Modern-day adjustment factors used to compare milk production of cows milked 2X to those milked 3X range from 12-14%, depending on the parity of the cow (VanRaden et al., 1999).

Much of the work on frequent milking for the entire lactation was conducted during the 1980s and 1990s, when there was great interest in switching milking regimes from 2X to 3X in order to increase milk production efficiency (Table 1). A typical response to 3X milking is illustrated in Figure 1, which shows the lactation curves of cows milked 2X or 3X for the entire lactation (re-drawn from Amos et al., 1985). Three-times daily milking increased milk production both at peak and through the entire lactation. Persistency of the lactation curve was also slightly increased, but this effect disappeared after approximately 180 DIM. In summaries of DHIA records, the increase in milk production with 3X was 13, 12, and 16% compared to 2X (Allen et al., 1986; Gisi et al., 1986; Smith et al., 2002). These reports were obtained from mostly Holstein herds, or combined Holstein and Jersey herds. Culotta and Schmidt (1988) suggested that smaller dairy breeds do not respond as well to frequent milking as larger breeds. Consistent with that hypothesis were observations of Campos et al. (1994), that relative to 2X, 3X increased milk production by 17.3 and 6.3% in Holsteins and Jerseys, respectively. In contrast, Copeland (1934) observed an impressive 21 and 19% increase in milk and fat production, respectively, from Jerseys milked 3X compared to those milked 2X. In addition, they reported a correlation (+.64) between the amount of milk cows produced prior to frequent milking and the magnitude of the response to frequent milking (Copeland, 1934). This led to speculation that higher producing cows better responded to increased milking frequency than did lower producing cows. The existence of such a relationship, however, has not been established. To the contrary, Erdman and Varner (1995) and Stockdale (2006) reviewed the literature on frequent milking, and reported no correlation between previous milk production and the response to changes in milking frequency. Instead, those researchers concluded that there was an incremental milk yield response. Relative to 2X, this fixed milk yield response was -6.2, +3.5 and +4.9 kg/d for once daily milking (1X), 3X, or 4X, respectively (Erdman & Varner, 1995). In agreement with this, Peel et al. (1979) estimated that the post-suckling increase in milk production of cows suckled for as little as one week during early lactation was 4.3 kg/d. They went so far as to provide the readers with the following equation to allow prediction of the milk yield response to various suckling regimes:

Increase in milk yield (%) = 4.± 0.(mean ±S.E.) x number of weeks suckled.E1
 Reference Parity Breed Duration of FM Change in milk yieldb (2X vs. 3Xc) Change in milk yield (2X vs. 4X) Riford, 1922 ≥ 1 Holstein; Guernsey Unknown + 4.6 kg/d Woodward, 1931 ≥ 1 Holstein Full lactation + 20% Copeland, 1934 ≥ 1 Jersey Through late lactation + 21% Rao and Ludri, 1984 ≥ 2 Brown Swiss x Sahiwal 50 to 130 DIMd + 1.34 kg/d + 1.73 kg/d DePeters et al., 1985 ≥ 2 Holstein Full lactation + 17% 1 Holstein Full lactation + 6% (NS) Amos et al., 1985 ≥ 2 Holstein Full lactation + 18.5% 1 Holstein Full lactation + 25.2% Allen et al., 1986 ≥ 2 Holstein Full lactation + 13.4% 1 Holstein Full lactation + 19.4% Gisi et al., 1986 ≥ 2 Holstein Full lactation + 12% 1 Holstein Full lactation + 14% Barnes et al., 1990 1 Holstein Full lactation + 14% Campos et al., 1994 1 Holstein Full lactation + 17.3% 1 Jersey Full lactation + 6.3% Klei et al., 1997 ≥ 1 Holstein Full lactation + 10.4% Smith et al., 2002 ≥ 1 Holstein Full lactation + 16%

### Table 1.

Summary of selected literature reports on the effects of frequent milking (FM) on milk yield of dairy cowsa.

Importantly, the increase in milk yield in response to frequent milking or suckling is not specific to dairy cows; milk production was increased with frequent milking or suckling of other dairy ruminants, including goats (Wilde et al., 1986), sheep (Geenty & Davison, 1982; Negrao et al., 2001; Nudda et al., 2002), and buffalo (Dash et al., 1976), as well as various cross breeds (Little et al., 1991; Krohn, 2001; Sidibe-Anago et al., 2008; Alvarez-Rodriguez et al., 2010).

### 1.2. Effects of frequent milking or suckling on milk composition and cow health

Reported effects of frequent milking on milk composition, somatic cell count (SCC) and herd health have been inconsistent. Many researchers have observed no effect of frequent milking on milk composition (Poole, 1982; Rao & Ludri, 1984; Amos et al., 1985; DePeters et al., 1985; Gisi et al., 1986), whereas some have observed a decrease in fat percentage (Allen et al., 1986; Smith et al., 2002). Due to the increase in milk production in response to frequent milking, however, there is often an increase in the total yield of fat and protein (Klei et al., 1997; Dahl et al., 2004b). With respect to SCC, some reports have indicated an association between frequent milking and decreased SCC, and these authors concluded that frequent milking may improve mammary health (Poole, 1982; Armstrong et al., 1985; Smith et al., 2002; Dahl et al., 2004b). Others have reported no effect of frequent milking on SCC (Waterman et al., 1983; DePeters et al., 1985; Gisi et al., 1986; Bar-Peled et al., 1995; Klei et al., 1997; Hale et al., 2003; Patton et al., 2006; Wall & McFadden, 2007a; Shields et al., 2011; Wright et al., 2011). The inconsistencies in the above reports may be the result of variations in timing and methods of sampling across experiments. Suckling of cows during early lactation has consistently been associated with a decrease in SCC and a decrease in the incidence of clinical mastitis, in some cases by up to 50% or more (Walsh, 1974; Edmunds, 1977; Little et al., 1991; Krohn, 2001). In fact, Walsh (1974) suggested that the increase in milk production elicited by suckling was probably due to the additional stimulus of the gland as well as the markedly improved mammary health of suckled animals.

Discrepancies also exist in the reported effects of frequent milking on reproductive performance and herd health. Some researchers have observed decreased reproductive performance in 3X cows compared to 2X cows (Ludwin, 1942; Armstrong et al., 1985; DePeters et al., 1985; Smith et al., 2002), whereas others have observed no effect (Poole, 1982; Amos et al., 1985; Gisi et al., 1986) or an improvement (Allen et al., 1986) in reproductive performance with 3X. Armstrong et al. (1985) suggested that any negative effects of frequent milking on herd health or reproductive performance may be associated with poor herd management. Well-controlled field studies using standardized sampling procedures will be necessary to verify the impact of milking frequency on milk composition, SCC and herd health, as well as identifying the interaction between frequent milking and herd management on these factors.

Similarly, suckling of a calf during early lactation of the cow is associated with increased weight loss and an increase in the days to first estrus (Margerison et al., 2002). In most cases, however, the delay in resumption of estrous cyclicity is offset by an increase in conception rate (Little et al., 1991; Krohn, 2001). Consequently, the effects of suckling on reproductive performance appear to be negligible, or in some cases positive (Table 2). In addition, Perez-Hernandez et al. (2002) reported that exposure of cows to a teaser bull decreased the effects of suckling on days to first estrus. Therefore, in agreement with the suggestion by Armstrong et al. (1985), simple changes in management - when feasible - can be used to overcome any potential negative impact of frequent milking or suckling regime on reproductive performance. Table 2 summarizes the findings of Little et al. (1991), who looked at lactation and reproductive performance of cows allowed to suckle calves for the first 90 days of lactation.

 Suckled & machine milked Machine milked Total milk production (kg) 1,894.10 1,644.60 Lactation length (d) 341 305 Dry matter feed intake (kg/d) 13.25 13.14 Days to estrus 101 41 Services per conception 1.3 1.9

### Table 2.

Productivity characteristics of crossbred cows subjected to suckling compared to machine-milked herdmates. Cows were allowed to suckle their calves for the first 90d of lactation and were also machine milked twice daily for the entire lactation (Little et al., 1991).

In addition to the increase in milk yield, suckling during early lactation has also been associated with an increase in lactation length (Everitt & Phillips, 1971; Little et al., 1991; Krohn, 2001). This is particularly relevant since some producers are interested in incorporating an extended lactation management scheme into their herd. Extended lactation offers the benefit of decreased health risk to the cow because there is less exposure to risk of metabolic disease, which highest during the transition period. This, in turn, decreases costs to the producer associated with treating sick cows and veterinary expenses. The use of frequent milking or suckling offers a tool for increasing lactation persistency during extended lactations. Indeed, Sorensen et al. (2008) reported that the use of increased milking frequency makes an extended lactation cycle economically viable for the producer, in addition to an observed improvement in the health of the cow.

### 1.3. Effects of suckling on calf health and performance

There are health benefits of keeping the calf with the cow and allowing the calf to suckle. Edmunds (1977) suggested that suckling calves generally perform better than those raised on milk replacer. Little et al. (1991) reported an increase in growth rate from birth to weaning of 0.2 kg/d for calves that were allowed to suckle vs. those that were raised on milk replacer. Similar observations were made by Alvarez-Rodríguez et al. (2010). In some cases, however, there is wide variation in the daily weight gain of suckling calves, especially if there are multiple calves on a single cow. Therefore, depending on the suckling scheme used, calves allowed to suckle don’t always have higher growth rates than calves raised on milk replacer (Krohn, 2001). In a review of the literature, Edmunds (1977) reported that suckling calves had superior daily weight gain, a decrease in scours, and minimal incidence of other diseases compared to bucket-reared herdmates. The author suggested that this was a result of decreased stress associated with separation from the cow, since suckling calves are allowed almost constant contact with their mothers.

An interesting study conducted by Bar-Peled et al. (1997) reported on the performance of first lactation heifers that were allowed to suckle as calves, and compared them to heifers that were reared on milk replacer. Their findings are summarized in table 3, and have major implications regarding the effects of suckling on performance as adults; suckling clearly improved production efficiency in several areas.

 Suckled Milk replacer Body weight at conception (kg) 358.6 327.2 Average daily gain (kg) 0.82 0.68 Age at conception (d) 394 426 Conception rate (%) 83.4 74.2 Calving age (d) 669 700 Milk production (kg/300d) 9624 9171

### Table 3.

Productivity characteristics of heifers that were allowed to suckle as calves compared to herdmates raised on milk replacer. Heifers were allowed to suckle as calves for the first six weeks of life. Herdmates were raised on milk replacer (Bar-Peled et al., 1997).

### 1.4. The effect of parity on the response to frequent milking

It has been reported that the milk production response to frequent milking was more pronounced in animals with smaller udder capacity, such as heifers (Woodward, 1931; Copeland, 1934; Lush & Shrode, 1953). Copeland (1934) speculated that this may be a function of increased udder pressure associated with less udder capacity. In agreement with this, Allen et al. (1986) observed that relative to 2X, 3X increased milk production by 13.4 and 19.4% in cows and heifers, respectively. In the previous year, however, another group reported that mature cows respond better to frequent milking than primiparous cows (DePeters et al., 1985). Of the studies that measured the milk yield response in heifers separately, most reported that heifers responded to frequent milking similarly to or better than multiparous cows. Therefore, no relationship has been established between parity and the magnitude of the milk yield response to frequent milking. To the contrary, we observed that that relative to multiparous cows, heifers respond similarly to increased milking frequency during early lactation (Wright et al., 2011). Similar observations have been made on the response of primi- vs. multi-parous cows to suckling during early lactation (Everitt & Phillips, 1971).

### 1.5. Economic impact of frequent milking or suckling

Several research groups have characterized the economic impact of frequent milking. Factors contributing to the profitability of frequent milking were labor, herd size, herd health, management, feed costs, and milk price (Armstrong et al., 1985; Culotta & Schmidt, 1988; Maltz et al., 2003). Rao and Ludri (1984) reported that 3X increased net income by 21% relative to 2X. More recently, we estimated a net increase of approximately $93/cow/yr when cows were milked 4X for the first 3 wk of lactation and milked 2X thereafter (Table 4; Wall & McFadden, 2007b). With respect to suckling, McKusick et al. (2001) estimated an increase in net income of$25 per ewe/suckling lamb pair in a mixed rearing system (in which ewes are suckled andmachine milked until lambs are weaned) relative to ewes solely machine milked and lambs raised on milk replacer. The increase in profitability of suckling management systems comes from eliminating the expenses associated with purchase of milk replacer, and also the increase in milk production of the dams after the suckling period. This, combined with the observation of Bar-Peled et al. (1997) that suckling calves perform better as adults, indicates multiple areas of economic gain with a suckling regime. Therefore, when there are no negative effects on animal health or reproductive performance, frequent milking or suckling has the potential to be a very profitable management tool.

## 2. Frequent milking or suckling during early lactation: A window of opportunity

An exciting development for both dairy producers and dairy scientists was the finding that the timing of implementation can influence the milk yield response to frequent milking. During middle and late lactation, frequent milking increased milk production; however cessation of frequent milking resulted in an immediate decrease in milk yield to pre-treatment levels (Elliott, 1961; Morag, 1973a, b; Svennersten et al., 1990). During early lactation, however, frequent milking for a short duration can stimulate milk production and the effect persists through the remainder of lactation, even after less frequent milking is resumed. This effect was originally observed in experiments designed to determine the milk yield loss associated with the use of nurse cows. Using identical twin cows, Everitt and Phillips (1971) discovered that suckling by calves in addition to machine milking during the first 8 to 10 weeks of lactation was associated with increased milk production after weaning and throughout the remainder of lactation in both primi- and multiparous cows (Figure 2). Shortly after this report, similar observations were made in both cows (Edmunds, 1977; Moss & O'Grady, 1978; Thomas et al., 1978; Fulkerson, 1981) and heifers (Fulkerson et al., 1978; Peel et al., 1979). Pearson et al. (1979) assigned cows to 3X for the first 143 d of lactation, followed by 2X thereafter. Although they did not report the full lactation curves, they measured milk yield for the entire lactation and reported that relative to 2X, cows that were milked 3X for the first 143 d of lactation produced more milk through 280 DIM (Pearson et al., 1979). Subsequently, it has been observed in numerous experiments that frequent milking during early lactation was associated with both acute and persistent increases in milk production (Table 5). These findings presented an opportunity for dairy producers; that an initial investment in labor could increase milk production efficiency for the remainder of lactation. Poole (1982) speculated that the practice might not be adopted, however, because producers would be discouraged by the partial decrease in milk production upon cessation of frequent milking, despite the significant carry-over effect.

 Milking routineb Feed cost during FMc Feed cost after FMd Labore Miscellaneous costf Extra milk income/cow/yrg Total net income/cow/yr 100-cow herdh 4X 1 to 21 $19.28$98.00 $84.00$0.50 $294.75$92.94 $9,293.60 4X 1 to 14$12.86 $100.75$56.00 $0.34$210.29 $40.33$4,032.85 4X 7 to 21 $12.86$98.00 $56.00$0.34 $239.33$72.12 \$7,212.35

### Table 4.

Potential economic return of milking four-times daily during early lactation.a

 Reference Treatment durationd Parity Change in milk yield (kg/d; 2X vs. 3X) Change in milk yield (kg/d; 2X vs. 4X) Change in milk yield (kg/d; 3X vs. 6X) Pearson et al., 1979 1 to 150 ≥ 2 + 2.2 Poole, 1982 1 to 140 ≥ 2 + 4.4 (acute) + 1.84 (pers.) 1 + 2.17 (acute) + .65 (pers.) Bar-Peled et al., 1995 1 to 42 ≥ 2 + 7.3 (acute) + 5.1 (pers.) Sanders, 2000 1 to 42 ≥ 2 + 6.0 (acute) + 3.7 (pers.) 1 + 1.7 (acute) Hale et al., 2003 1 to 21 ≥ 2 + 8.6 (acute) + 2.6 (pers.) Dahl et al., 2004b 1 to 21 ≥ 2 + 14 (acute) + 3.7 (full lactation)

### Table 5.

Summary of select literature reports on the effects of frequent milking during early lactation on milk yield of dairy cowsa

In an attempt to minimize additional costs associated with frequent milking, and to investigate the response of dairy cows to frequent milking or suckling during a short interval of time in early lactation, Bar-Peled et al. (1995) compared 3X to 6X or 3X+suckling for the first 6 wk of lactation, followed by 3X of all cows. Relative to cows milked 3X during the entire lactation, 6X and 3X + suckling acutely increased milk production by 7.3 and 14.7 kg/d, respectively (Bar-Peled et al., 1995). Cessation of frequent milking or suckling was associated with a partial decline in milk production; however, a carry-over effect was observed in 6X cows (+5.1 kg/d relative to 3X; Bar-Peled et al., 1995). In a similar experiment, Sanders et al. (2000) observed an acute increase of 6 kg/d and a carry over response of 3.7 kg/d in 6X cows relative to 3X cows. In heifers, the acute response to 6X was lower in magnitude (+1.7 kg/d), and no carry over effect was observed (Sanders et al., 2000).

A summary of literature reports on the effect of suckling during early lactation on milk production is presented in Table 6. The results of subsequent experiments have further narrowed down the ‘window’ during early lactation wherein frequent milking can increase milk production for the remainder of lactation. Hale et al. (2003) assigned cows to 2X or to 4X for the first 3 wk of lactation, followed by 2X. Four times daily milking was associated with an acute increase of 8.8 kg/d and a carry over effect of 2.6 kg/d for the remainder of lactation. A treatment interval of 1 to 21 DIM was also used in a field study by Dahl et al. (2004b), who observed similar effects of frequent milking during early lactation. In contrast, VanBaale et al. (2005) assigned cows to 3X or 6X for the first 7, 14, or 21 d of lactation and reported that 6X did not increase milk production relative to 3X. Their observations were inconsistent with previous reports, and the authors speculated that facility logistics may have influenced their results because 6X cows were housed farther away from the milking parlor and spent a considerably longer time away from their pen than 3X cows (VanBaale et al., 2005). With the exception of one negative report (VanBaale et al., 2005), and one abstract (Fernandez et al., 2004), it is generally accepted that frequent milking increases milk yield, and that frequent milking or suckling during early lactation can increase milk production for the remainder of lactation (see Tables 5 and 6). The mechanistic basis for the milk yield response to frequent milking, however, is poorly understood. Even less understood are the mechanisms involved in the persistent effect on milk yield of frequent milking during early lactation.

 Reference Treatment Duration (DIM) Parity 2X vs. 1X + suckle 2X vs. 2X + suckle 3X vs. 3X + suckle Everitt and Phillips, 1971 1 to 70 2+ (+) 1.2 kg/d (pers.a) 1 (+) .87 kg/d (pers.) Walsh, 1974 1 to 100 2+ (+) 11.3% (acuteb) (+) 7.7% (pers.) Moss and O'Grady, 1978 1 to 56 2+ (+) 3.3 kg/d (no pers.) Thomas et al., 1978 1 to 56 2+ (+) 1.68 kg/d (acute) (+) 1.77 kg/d (pers.) Fulkerson et al., 1978 1 to 56 1 (+) 16% (full lactation) Pearson et al., 1979 1 to 150 2+ Peel et al., 1979 1 to 28 1 (+) 13% (pers.) Fulkerson et al., 1981 1 to 7 2+ (+) 9% (pers.) Teeluck et al., 1981 1 to 90 2+ (+) 2.2 kg/d (+) 3.33 kg/d Poole, 1982 1 to 140 2+ Little et al., 1991 1 to 94 5 (+) 15% (full lactation) Bar-Peled et al., 1995 1 to 42 2+ + 14.7 kg/d (acute)

### Table 6.

Summary of select literature reports on the effects of suckling during early lactation on milk yield of dairy cows

## 3. Local regulation of the milk yield response to frequent milking

Shortly after the report in the 1930s that frequent milking increased milk production, several studies utilized unilateral frequent milking (UFM) to investigate the effect. Half-udder designs are extremely powerful because they eliminate variation between animals due to environment, nutrition, and genetics. In addition, both udder halves are theoretically exposed to the same systemic factors, hence responses to frequent milking are strictly at the level of the mammary gland. A summary of the milk yield response to frequent milking in selected half-udder experiments is presented in Table 7. The early reports provided strong evidence for local regulation of milk production, and increases in milk yield from 8.4 to 32% in the frequently-milked udder half were observed (Ludwick et al., 1941; Cash & Yapp, 1950; Agarwala & Sundaresan, 1955; Claesson et al., 1959). Morag (1973b) reported that the increase in milk production in response to UFM occurs within 24 h, and the magnitude of the response was independent of previous milk production. In addition, heifers respond to UFM; Hillerton et al. (1990) milked udder halves 2X or 4X for 4 wk during mid-lactation. In both cows and heifers, milk production of 4X udder halves increased by 10.4% relative to 2X udder halves (Hillerton et al., 1990).

 Reference Stage of Lactation Duration of UFM Parity Change in milk yield(%; 2X vs. 3X) Change in milk yield (2X vs. 4X) Cash & Yapp, 1950 - Full lactation ≥ 2 + 32 Agarwala & Sundaresan, 1955 Early 25 d intervals ≥ 2 + 8.4 Hillerton et al., 1990 Mid-late 28 d ≥ 1 (+) 10.4% Knight et al., 1992 Mid 14 d 1 (+) 14% Knight, 1992 Early 42 d ≥ 1 + 10.4 Norgaard et al., 2005 Mid 7 d ≥ 1 (+) 18% Wall & McFadden, 2007a 1 DIMd 21 d ≥ 2 (+) 3.5 kg/d (acutee) (+) 1.8 kg/d (pers.f) Wall & McFadden, 2007b 1 DIM 14 d ≥ 2 (+) 3.7 kg/d (acute) (+) 1.2 kg/d (pers.) 7 DIM 14 d ≥ 2 (+) 2.9 kg/d (acute) (+) 1.5 kg/d (pers.) Wright et al., 2011 1 DIM 21 d 1 (+) 2.8 kg/d (acutee) (+) 0.9 kg/d (pers.) Shields et al., 2011 1 DIM 21 d ≥ 2 (+) 3.4 kg/d

### Table 7.

Summary of select literature reports on the effects of unilateral frequent milking (UFM) on milk yielda

As mentioned previously, an emerging theme in these experiments has been that the effects of frequent milking during early lactation on milk production persist even after a lower milking frequency is resumed (Bar-Peled et al., 1995; Hale et al., 2003; Dahl et al., 2004b). Although this persistent milk yield response has been consistently observed (Table 5), it was unknown whether the response was regulated by hormones, by local factors within the mammary gland, or by the combination of the two. To investigate this question, we used a half-udder model and assigned cows to UFM (4X of the right udder half, 2X of the left udder half) for d 1 to 21 of lactation, followed by 2X for the remainder of lactation (Wall & McFadden, 2007a). When the half-udder milk yields were adjusted to the equivalent of a whole udder basis, the acute and long-term milk yield responses to frequent milking that we observed were consistent with those reported by Hale et al. (2003). Therefore, our results indicated that both the acute and persistent effects of frequent milking during early lactation are regulated by local factors within the mammary gland. This is illustrated in Figure 3A and B. Figure 3A (re-drawn from Bar-Peled et al., 1995) shows the milk yield response of multiparous cows to 6X for the first 6 wk of lactation, followed by 3X. We observed a similar effect using a half-udder experiment (Figure 3B), and the milk yield response lasted through 270 DIM. This finding presents some intriguing questions and research opportunities. First, what are the local factors that regulate milk production capacity of the mammary gland? Once the factor(s) have been identified and pathways understood, how can we refine our approach to maximize milk production efficiency of dairy cows? Now that it is established that the factors are indeed local, the problem has become relatively simplified. Extremely powerful, within cow experiments that are less sensitive to the influence of environment, genetics and nutrition can now be designed to ask such mechanistic questions.

On the road to refinement, one theme that has transpired is the existence of a ‘window’ of time wherein the mammary gland is especially responsive to frequent milking. The duration of this window has been shortened from the first 10 wk of lactation (Moss & O'Grady, 1978; Thomas et al., 1978) to the first 6 wk of lactation (Bar-Peled et al., 1995; Sanders et al., 2000), and shortened further still to the first 3 wk of lactation (Hale et al., 2003; Dahl et al., 2004b; Wall & McFadden, 2007a). It was unknown whether a shorter duration or altered timing of frequent milking during early lactation would still elicit a persistent effect on milk production; however, since any costs associated with extra labor are increased only during frequent milking, it was of great interest to shorten the duration of frequent milking if a persistent increase in milk yield could still be observed. To answer this question, we assigned cows to UFM (4X of the right udder half, 2X of the left udder half) for d 1 to 14 or d 7 to 21 of lactation (Wall & McFadden, 2007b). We observed an acute milk yield response in both treatments; and a significant carry-over effect in the d 7 to 21 group. There was a numerical carry-over for the d 1 to 14 group; however it was not significant. Our results indicate that within the first 21 DIM, an interval of frequent milking as short as 2 wk can elicit a persistent increase in milk production. As mentioned previously, similar observations have been made on the response of cows and heifers to suckling during early lactation (Fulkerson et al., 1978; Peel et al., 1979). Further narrowing of this “window” within the first 21 of lactation, as well as characterization of the cellular response could provide insight into the mechanisms underlying the receptiveness of the mammary gland to stimulus during this time.

## 4. Endocrine response to frequent milking or suckling

It has long been thought that the hormones released at milking may be involved in regulating the galactopoietic effects of frequent milking on milk production. Indeed, multiple hormones are released during milking including glucocorticoids, oxytocin, and prolactin (Tucker et al., 1975; Carruthers & Hafs, 1980; Akers & Lefcourt, 1982). Oxytocin is responsible for milk ejection. Cows suckling calves are thought to have more efficient milk ejection due to increased secretion of oxytocin elicited by the presence of the calf. In fact, on dairies using cross breeds and cows not bred for high milk production, the calf is often used as a facilitator of milk letdown during milkings (Little et al., 1991; Krohn, 2001; Bruckmaier & Wellnitz, 2008). In addition, treatment with exogenous oxytocin was associated with increased milk production of both dairy cows (Nostrand et al., 1991; Ballou et al., 1993; Lollivier & Marnet, 2005) and sheep (Zamiri et al., 2001). Therefore, it is possible that oxytocin is involved in regulating the increase in milk production elicited by frequent milking or suckling, perhaps by allowing for more complete milk removal and a decrease in negative feedback on the gland.

Along with enhanced milk production, Bar-Peled et al. (1995) observed increased concentrations of growth hormone, insulin-like growth factor-1, oxytocin and prolactin in circulation of cows that were frequently milked or suckled. In addition, the magnitude of milking-induced PRL release declines concomitantly with the decrease in milk production as lactation progresses (Koprowski & Tucker, 1973). Consequently, PRL has been hypothesized as a candidate regulator of the effects of frequent milking on milk production (Dahl et al., 2004a). In an attempt to determine whether milking-induced PRL release indeed mediates the effects of frequent milking on milk production, we assigned cows to 2X, 4X, or 2X + twice daily injections of PRL (Crawford et al., 2004; Wall et al., 2006). Four times daily milking or PRL injections increased milk production relative to 2X (Crawford et al., 2004); however our results indicated that PRL injection or frequent milking exerted distinct effects on mammary cell growth and gene expression, thus probably increased milk production via separate mechanisms (Wall et al., 2006). The response to unilateral frequent milking during early lactation supports this concept; frequent milking may stimulate milk production via local factors, whereas PRL injections may increase milk yield through a more systemic pathway.

## 5. Cellular response to frequent milking

Several authors have speculated that frequent milking increases milk yield via an increase in mammary cell number and (or) activity (Bar-Peled et al., 1995; Stelwagen & Knight, 1997; Sanders et al., 2000; Hale et al., 2003), both of which are critical to improved lactation performance (Capuco et al., 2003). Hillerton et al. (1990) observed an increase in activity of mammary enzymes, protein and lactose synthesis (in heifers only), DNA synthesis, and alveolar area in response to increased milking frequency, and concluded that cellular differentiation and proliferation were optimized with frequent milking. Hale et al. (2003) reported an increase in mammary cell proliferation at 7 DIM in cows that were milked 4X for the first 3 wk of lactation compared to cows milked 2X; however, differences in proliferation were only observed in one of the two frequently milked cow groups. In contrast to those experiments, Norgaard et al. (2005) reported that despite an increase in milk yield (+18%), there was no effect of frequent milking on cell death, proliferation, or enzyme activities in the mammary gland. In agreement with that report, we have observed across multiple experiments that relative to 2X, 4X did not affect mammary cell proliferation or apoptosis (Wall et al., 2006; Wall & McFadden, 2010; Wall et al., 2011b), indicating that changes in milking frequency influence milk yield through an alternative mechanism.

Using a unilateral frequent milking model and a functional genomics approach, we determined that the increase in milk yield associated with frequent milking is regulated by changes in gene expression elicited by removal of milk from the gland (Wall et al., 2011a). We then used a sequential biopsy approach and obtained mammary tissue at various times during and after exposure to unilateral frequent milking and determined that the temporal expression of 64 genes was co-regulated by unilateral frequent milking (Wall et al., 2011b). Importantly, the pattern of differential expression of the 64 genes was negatively correlated with differential milk yield (Figure 4); therefore, we hypothesize that we have identified a pathway for the autocrine regulation of milk production. Furthermore, this transcriptional signature appears to be malleable and adaptable to the needs of the offspring (mimicked by changes in milking frequency), since expression of some of the genes was still different between udder halves nearly three weeks after cessation of treatment (Figure 4). Future experiments will clarify the role of these genes in the mammary gland and their involvement in the autocrine regulation of milk production.

What is unique to early lactation, when the stimulus of frequent milking for a short duration can elicit a persistent increase in milk production? This question remains unanswered, but work by Stelwagen and Knight (1997) has provided some clues. Using a half-udder model, they compared 1X to 2X of cows in early or late lactation and reported a more dramatic increase in milk secretion efficiency in response to 2X during early lactation compared to late lactation (Stelwagen & Knight, 1997). In agreement, Walsh (1974) observed different effects of suckling during early vs. late lactation on mammary health. During early lactation, suckling of a calf was associated with a 27% decrease in clinical mastitis, whereas suckling during late lactation had no effect (Walsh, 1974). Taken together, the observations of Walsh (1974) and Stelwagen and Knight (1997) indicate that there are distinct differences in the cell population during early vs. late lactation. It is possible that during early lactation, there are more secretory cells present in the mammary gland, and these cells may have more potential to respond to stimulus than cells present in late lactation. Frequent milking may prevent otherwise unused cells from undergoing apoptosis, or may provide the stimulus to push the cells to reach higher levels of differentiation and secretory capacity. These scenarios could result in an increase in the number of cells in the gland throughout lactation, an increase in the activity of cells throughout the lactation, or both. Shorten et al. (2002) proposed a hypothetical model by which frequent milking for the entire lactation increases the number of active alveoli by reducing the rates of quiescence and senescence in the mammary gland. If such an event occurs with frequent milking during early lactation, this could permanently increase the number of actively secreting alveoli and enhance milk production potential for the remainder of lactation. Many of the biopsy studies that have been previously conducted could have captured changes in mammary cell activity, but would not have captured changes in total cell number or in rates of quiescence and senescence within the gland.

## 6. Conclusions

Research in the area of frequent milking of dairy cows has established a robust milk yield response to increased milking frequency or suckling, and has identified a window of time during early lactation wherein the mammary gland is especially responsive to the stimulus of frequent milk removal. In addition, there is now evidence that this response is regulated within the mammary gland. Consequently, the concept of ‘use it or lose it’ is becoming more clearly established, that is, the stimulus of frequent milking or suckling during early lactation permanently increases the milk production capacity of the mammary gland. Exciting research opportunities now present themselves, and ongoing experiments seek to identify the local factor(s) that are involved in the regulation of milk production efficiency of dairy cows. The opportunity now exists for dairy scientists to identify the mechanisms involved in local regulation of milk production potential, and for dairy producers to further refine milking management practices to maximize milk production efficiency of their operations.

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# Clear and not so big 3. (10 points) Prove that if a > 0, then there...

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clear and not so big
3. (10 points) Prove that if a > 0, then there exists n E N such that ! <a<n.

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Table 3: Chi-Square_Tests Value 421.6491 428.636 27.655 Asymptotic Significance (2-sided) 0oo 0oo 00o Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association Nof Valid Cases 23110 a. 0 cells (0.0%) have expected count Iess than 5 The minimum expected count is 353.91. Table 4: Symmetric_Meas...
##### Prove from Definition 4.5 .4 that if $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ is linearly independent and if $A$ is an invertible $n \times n$ matrix, then the set $\left\{A \mathbf{v}_{1}, A \mathbf{v}_{2}, \ldots, A \mathbf{v}_{n}\right\}$ is linearly independent.
Prove from Definition 4.5 .4 that if $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ is linearly independent and if $A$ is an invertible $n \times n$ matrix, then the set $\left\{A \mathbf{v}_{1}, A \mathbf{v}_{2}, \ldots, A \mathbf{v}_{n}\right\}$ is linearly independent....
##### 2. For the DFG of the following filter, find the required DFG for r(l) = 0,r(2) = 0, r(3) = -L,r(4) =-2,r(5) = -1,r(6) = 0, r(7) = 0,r(8) = 0.INOUT
2. For the DFG of the following filter, find the required DFG for r(l) = 0,r(2) = 0, r(3) = -L,r(4) =-2,r(5) = -1,r(6) = 0, r(7) = 0,r(8) = 0. IN OUT...
##### Find the magnitude and direction of the resultant of the two forces shown knowing that $P=400 \mathrm{N}$ and $Q=300 \mathrm{N}$.
Find the magnitude and direction of the resultant of the two forces shown knowing that $P=400 \mathrm{N}$ and $Q=300 \mathrm{N}$....
##### Question 3:(5 marks) The balances on December 31, 2017 are available for Denver Company: $2,550 Accounts... Question 3:(5 marks) The balances on December 31, 2017 are available for Denver Company:$2,550 Accounts payable 3,550 Accounts receivable 6,250 Accumulated depreciation Retained earnings, December 31, 2017 2,300 Cash $2.300 Cost of goods sold 2,500 Depreciation expense 8,800 Dividends declared 29,0... 1 answer ##### Find the interest paid on a loan of$2,200 for two years at a simple interest...
Find the interest paid on a loan of $2,200 for two years at a simple interest rate of 11% per year. The interest on the loan is$17...
##### In order to sequence a strand of DNA, all four dNTPs, ddGTP, DNA polymerase and primers...
In order to sequence a strand of DNA, all four dNTPs, ddGTP, DNA polymerase and primers (5’ ACTGA 3’) are added to a test tube. A template strand of DNA is shown below. How many fragments of different lengths would you have at the end of the sequencing process? (write down the sequence o...
##### Problem 6. (30 points). The diameters of each solid segment are dab=0.75 in., d3c= 1.5 in.,...
Problem 6. (30 points). The diameters of each solid segment are dab=0.75 in., d3c= 1.5 in., and dcp=0.5 in. Draw all appropriate free body diagrams and determine the values average normal stresses at points E and F. Indicate whether Tension or Compression. 5 kip 2 kip 8 kip 6 kip A D 5 kip B C 2 kip...
##### The circumference of the circle is 24cm what is the radius
the circumference of the circle is 24cmwhat is the radius?24=2 pi r24/3.14=88/2=44cm...
##### 7. For the graph of g(x) shown at right, (^ & B: 3 pts ca) At what x-value(s) does g'(x) change its sign?B) For what interval(s) ofxis both g"(x) > 0 and g'(x) < 0 ?C) (4 pts) Sketch the derivative; g on the same axesIfyou are working in your nolebook; label the x-values on your grph:
7. For the graph of g(x) shown at right, (^ & B: 3 pts ca) At what x-value(s) does g'(x) change its sign? B) For what interval(s) ofxis both g"(x) > 0 and g'(x) < 0 ? C) (4 pts) Sketch the derivative; g on the same axes Ifyou are working in your nolebook; label the x-values...
##### Point) Consider the surface F(z,y,2) = x622 + sin(y+22) + 1 = 0. Find the following partial derivatives Dz 8y
point) Consider the surface F(z,y,2) = x622 + sin(y+22) + 1 = 0. Find the following partial derivatives Dz 8y...
##### Energy 9
Truck suspensions often have "helper springs" that engage at highloads. One such arrangement is a leaf spring with a helper coilspring mounted on the axle, as shown inFigure P5.22. When the mainleaf spring is compressed by distance y0, thehelper spring engages and then helps to support any a...
##### 1. Find the frequency of  rudio wave with wavelength ^ 200 m considering that it truvels with peed 0= 3 * 10"ms;
1. Find the frequency of # rudio wave with wavelength ^ 200 m considering that it truvels with peed 0= 3 * 10"ms;...
##### Points) Negate the following statement:(P xor 9) -rChoose the correct statement: XOT q)v ((p^ 9) v (~P^ 9))^ P^9) ^ (p^ @))^ ~Te (P xor q) ^r
points) Negate the following statement: (P xor 9) -r Choose the correct statement: XOT q)v ((p^ 9) v (~P^ 9))^ P^9) ^ (p^ @))^ ~Te (P xor q) ^r...
##### Eouslbn HepFSsume matwhen aduhssmanphcnesancomly a6laced-nemin IeatinnsS5caojlismanchonrandomiy selecled fndCOaltalyIF [rbrnbnolhemcheir emanphonas Inerunce 0T Drnaaprobabaity (Type nlegs Dac731 [ounted Iolr Cacmal pleces neajed
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##### 3372 1. J7 dx (Use Partial factions) 36-9
3372 1. J7 dx (Use Partial factions) 36-9...
##### -/10 POINTSScalccca 8.2.047.Mi;MY NOTESAssume the nth partlalGeriesthe follawing(a) Fina a Ior @(D) Find-/10 POINTSSCALCCCA 8.2.048,MY NOTESASSUMe theGe-lesfollawlng50 = 2 ns-1(a) Find & (0(b) FindpJua
-/10 POINTS Scalccca 8.2.047.Mi; MY NOTES Assume the nth partlal Geries the follawing (a) Fina a Ior @ (D) Find -/10 POINTS SCALCCCA 8.2.048, MY NOTES ASSUMe the Ge-les follawlng 50 = 2 ns-1 (a) Find & (0 (b) Find pJua...
##### Q1. Calculate AH", 4S", and AG" for the reaction given below at 259C. Is the standard reaction given below spontaneous at this temperature (25 %C) (10 P): HzOza) HzOa) 0zg) AG' rxn AH" rxn TAS" rxnH2O2a)HOa)0zg) 0.O0 205.2AH" kllmol S" Jlmol K-187.78 109.60285.83 70.00
Q1. Calculate AH", 4S", and AG" for the reaction given below at 259C. Is the standard reaction given below spontaneous at this temperature (25 %C) (10 P): HzOza) HzOa) 0zg) AG' rxn AH" rxn TAS" rxn H2O2a) HOa) 0zg) 0.O0 205.2 AH" kllmol S" Jlmol K -187.78 109....
##### Use the References to access important values if needed for this question. A mixture of neon...
Use the References to access important values if needed for this question. A mixture of neon and krypton gases, at a total pressure of 950 mm Hg, contains 4.90 grams of neon and 10.4 grams of krypton. What is the partial pressure of each gas in the mixture? mm Hg PNe - Pkr = mm Hg Submit Answer Retr...
##### Loca bank using Winters' method with 0.2_ 0.1_ and Y = 0_ t0 forecast the number of customers served each day. The bank open Monday through Fnday: At the end of the previous week, the following seasona indexes have been estimated: Monday; 0.95; Tuesday; 0.90; Wednesday; .80; Thursday; 1.25; Friday; 1.10. Also , the current estimates of level and trend are 20 and After observing that 30 customers are served by the bank on this Monday; forecast the number customers who will be served on each
loca bank using Winters' method with 0.2_ 0.1_ and Y = 0_ t0 forecast the number of customers served each day. The bank open Monday through Fnday: At the end of the previous week, the following seasona indexes have been estimated: Monday; 0.95; Tuesday; 0.90; Wednesday; .80; Thursday; 1.25; Fri...
##### Dit View History Bookmarks Window Help Chapter 6 Homework Homework 6 Saved Select characteristics of biofilms...
dit View History Bookmarks Window Help Chapter 6 Homework Homework 6 Saved Select characteristics of biofilms that differentiate them from independent-living bacteria. Check All That Apply Microorganisms live singly, independently Microorganisms participate in quorum sensing Microorganisms secrete e...
Problem 3-60 Extensions of the CVP Basic Model-Multiple Products and Taxes (LO 3-4) Assume that Ocean King Products sells three varieties of canned seafood with the following prices and costs: Selling Price per Case Variable Cost per Case Fixed Cost per Month Variety 1 $16$ 13 – ...

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 Order related concepts for arbitrary groupoids Bull. Korean Math. Soc. 2017 Vol. 54, No. 4, 1373-1386 https://doi.org/10.4134/BKMS.b160593Published online July 31, 2017 Hee Sik Kim, Joseph Neggers, and Keum Sook So Hanyang University, University of Alabama, Hallym University Abstract : In this paper, we introduce and explore suggested notions of above', below' and `between' in general groupoids, $Bin(X)$, as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), $d/BCK$-algebras, linear groupoids over fields and special cases, in order to illustrate the usefulness of these ideas. Additionally, for groupoid-classes (e.g., $BCK$-algebras) where these notions have already been accepted in a standard form, we look at connections between the several definitions which result from our introduction of these ideas as presented in this paper. Keywords : below, above, between, $d/BCK$-algebra, poset, $Bin(X)$, order-preserving, $\beta$-medial MSC numbers : Primary 20N02, 20M10, 06A06 Downloads: Full-text PDF


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# Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - A)^{-1}$.

Is there a spectral theorem for this context, which gives sufficient conditions on $A$ for the matrix $H$ to be diagonalizable? Are there sufficient conditions on $A$ to guarantee that the eigenvalues of $H$ decay rapidly (e.g., exponentially)?

I am sure that such questions have been analyzed in the past, perhaps in the economics literature. However, I have been unable to find a reference. While I phrased these question in terms of matrices, I of course would also be interested in the more general context of linear operators on Hilbert spaces.

-

If you take a Schur form $A=QTQ^T$, then $H=QT(I-T)^{-1}Q^T$, and you can ignore the orthogonal factors $Q$. You might also want to set $N=I-T$, so that $Q^THQ=N^{-1}-I$. Now the problem looks much simpler.
• $H$ diagonalizable $\Leftrightarrow$ $N^{-1}$ diagonalizable $\Leftrightarrow$ $N$ diagonalizable $\Leftrightarrow$ $T$ diagonalizable $\Leftrightarrow$ $A$ diagonalizable
• the eigenvalues of $H$ are $\frac{\lambda_i}{1-\lambda_i}$, where $\lambda_i$ are the eigenvalues of $A$ (you can read it off $H=QT(I-T)^{-1}Q^T$). So they decay to zero if the eigenvalues of $A$ do so.
Some pointers: the first chapter of Higham's book Functions of matrices treats with much general cases, and what you are doing is applying a special Cayley transform to the matrix $A$.
@Federico Poloni: Tom LaGatta says $A$ is not symmetric. How does the orthogonal matrix $Q$ diagonalize $A$? If it can not, what is the relevance of tranforming $A$ to $T$? – Hans May 6 '13 at 19:24


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## définition - Bandwidth (signal processing)

voir la définition de Wikipedia

Wikipedia

# Bandwidth (signal processing)

Baseband bandwidth. Here the bandwidth equals the upper frequency.

Bandwidth is the difference between the upper and lower frequencies in a continuous set of frequencies. It is typically measured in hertz, and may sometimes refer to passband bandwidth, sometimes to baseband bandwidth, depending on context. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, an electronic filter, a communication channel, or a signal spectrum. In case of a low-pass filter or baseband signal, the bandwidth is equal to its upper cutoff frequency. The term baseband bandwidth always refers to the upper cutoff frequency, regardless of whether the filter is bandpass or low-pass.

Bandwidth in hertz is a central concept in many fields, including electronics, information theory, radio communications, signal processing, and spectroscopy. A key characteristic of bandwidth is that a band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum (assuming equivalent noise level). For example, a 5 kHz band can carry a telephone conversation whether that band is at baseband (as in your POTS telephone line) or modulated to some higher (passband) frequency.

In computer networking and other digital fields, the term bandwidth often refers to a data rate measured in bits per second, for example network throughput, sometimes denoted network bandwidth, data bandwidth or digital bandwidth. The reason is that according to Hartley's law, the digital data rate limit (or channel capacity) of a physical communication link is proportional to its bandwidth in hertz, sometimes denoted radio frequency (RF) bandwidth, signal bandwidth, frequency bandwidth, spectral bandwidth or analog bandwidth. For bandwidth as a computing term, less ambiguous terms are bit rate, throughput, maximum throughput, goodput or channel capacity.

## Overview

Bandwidth is a key concept in many telephony applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier wave, whereas in optics it is the width of an individual spectral line or the entire spectral range.

In many signal processing contexts, bandwidth is a valuable and limited resource. For example, an FM radio receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the United States) may apportion the regionally available bandwidth to broadcast license holders so that their signals do not mutually interfere. Each transmitter owns a slice of bandwidth, a valuable (if intangible) commodity.

For different applications there are different precise definitions. For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero. This would correspond to the mathematical notion of the support of a function (i.e., the total "length" of values for which the function is nonzero). A less strict and more practically useful definition will refer to the frequencies where the frequency function is small. Small could mean less than 3 dB below (i.e., power output < 1/2 or voltage output < $\scriptstyle \sqrt{1/2} \ \approx \$ 0.707 of) the maximum value, or more rarely 10 dB below, or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes.

Bandwidth typically refers to baseband bandwidth in the context of, for example, sampling theorem and Nyquist sampling rate, while it refers to passband bandwidth in the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems.

## X-dB bandwidth

A graph of a bandpass filter's gain magnitude, illustrating the concept of −3 dB bandwidth at a gain of 0.707. The frequency axis of this symbolic diagram can be linear or logarithmically scaled.

In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density is nonzero or above a small threshold value. That definition is used in calculations of the lowest sampling rate that will satisfy the sampling theorem. Because this range of non-zero amplitude may be very broad or infinite, this definition is typically relaxed so that the bandwidth is defined as the range of frequencies in which the signal's spectral density is above a certain threshold relative to its maximum. Most commonly, bandwidth refers to the 3-dB bandwidth, that is, the frequency range within which the spectral density (in W/Hz or V2/Hz) is above half its maximum value (or the spectral amplitude, in V or V/Hz, is more than 70.7% of its maximum); that is, above −3 dB relative to the peak.[1]

The word bandwidth applies to signals as described above, but it could also apply to systems, for example filters or communication channels. To say that a system has a certain bandwidth means that the system can process signals of that bandwidth, or that the system reduces the bandwidth of a white noise input to that bandwidth.

The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which in the passband filter case is typically at or near its center frequency, and in the lowpass filter is near 0 hertz. If the maximum gain is 0 dB, the 3 dB gain is the range where the gain is more than -3dB, or the attenuation is less than + 3dB. This is also the range of frequencies where the amplitude gain is above 70.7% of the maximum amplitude gain, and above half the maximum power gain. This same "half power gain" convention is also used in spectral width, and more generally for extent of functions as full width at half maximum (FWHM).

In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB +/- a small number of dB, for example within the +/- 1 dB interval. In the stopband(s), the required attenuation in dB is above a certain level, for example >100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1dB-bandwidth. If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.

A commonly used quantity is fractional bandwidth. This is the bandwidth of a device divided by its center frequency. E.g., a passband filter that has a bandwidth of 2 MHz with center frequency 10 MHz will have a fractional bandwidth of 2/10, or 20%.

In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley formula, the bandwidth refers to the frequency range within which the gain is non-zero, or the gain in dB is below a very large value.

The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth, since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as $B = 2W$, where $B$ is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and $W$ is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require a lowpass filter with cutoff frequency of at least $W$ to stay intact, and the physical passband channel would require a passband filter of at least $B$ to stay intact.

In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak.

In basic electric circuit theory, when studying band-pass and band-reject filters, the bandwidth represents the distance between the two points in the frequency domain where the signal is $\frac{1}{\sqrt{2}}$ of the maximum signal amplitude (half power).

## Antenna systems

In the field of antennas, two different methods of expressing relative bandwidth are used for narrowband and wideband antennas.[2] For either, a set of criteria is established to define the extents of the bandwidth, such as input impedance, pattern, or polarization.

Percent bandwidth, usually used for narrowband antennas, is used defined as $%B=\frac{f_H-f_L}{f_c}=2 \frac{f_H-f_L}{f_H+f_L}$. The theoretical limit to percent bandwidth is 200%, which occurs for $f_L = 0$.

Fractional bandwidth or Ratio bandwidth, usually used for wideband antennas, is defined as $B = f_H/f_L,$ and is typically presented in the form of $B:1$. Fractional bandwidth is used for wideband antennas because of the compression of the percent bandwidth that occurs mathematically with percent bandwidths above 100%, which corresponds to a fractional bandwidth of 3:1.

$\mathrm{If}\ %B =2 \frac{f_H-f_L}{f_H+f_L} = p%,\ B = \frac{200+p}{200-p}$.

## Photonics

In photonics, the term bandwidth occurs in a variety of meanings:

• the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
• the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
• the gain bandwidth of an optical amplifier
• the width of the range of some other phenomenon (e.g., a reflection, the phase matching of a nonlinear process, or some resonance)
• the maximum modulation frequency (or range of modulation frequencies) of an optical modulator
• the range of frequencies in which some measurement apparatus (e.g., a powermeter) can operate
• the data rate (e.g., in Gbit/s) achieved in an optical communication system; see bandwidth (computing).

A related concept is the spectral linewidth of the radiation emitted by excited atoms.

## References

1. ^ Van Valkenburg, M. E.. Network Analysis (3rd edition ed.). pp. 383–384. ISBN 0-13-611095-9. Retrieved 2008-06-22.
2. ^ Stutzman, Warren L., and Gary A. Theiele. Antenna Theory and Design. 2nd Ed. New York: 1998. ISBN 0-471-02590-9

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