Shannon's sampling theorem states that a band-limited signal with maximum frequency fm" role="presentation" style="position: relative;">fmfmf\_m can be reconstructed from its samples taken at a rate fs" role="presentation" style="position: relative;">fsfsf\_s such that fs&gt;2fm" role="presentation" style="position: relative;">fs>2fmfs>2fmf\_s > 2f\_m. In the case of sampling with an ideal sampling function (Dirac comb), the spectrum of the sampled signal can be derived as follows:fm" role="presentation" style="position: relative;">fmfmf\_m can be reconstructed from its samples taken at a rate fs" role="presentation" style="position: relative;">fsfsf\_s such that fs&gt;2fm" role="presentation" style="position: relative;">fs>2fmfs>2fmf\_s > 2f\_m. In the case of sampling with an ideal sampling function (Dirac comb), the spectrum offm" role="presentation" style="position: relative;">fmfmf\_m can be reconstructed from its samples taken at a

Let the continuous-time signal be xc(t)" role="presentation" style="position: relative;">xc(t)xc(t)x\_c(t) with Fourier transform Xc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), and let the sampled signal be xs(t)=xc(t)p(t)" role="presentation" style="position: relative;">xs(t)=xc(t)p(t)xs(t)=xc(t)p(t)x\_s(t) = x\_c(t)p(t), where p(t)" role="presentation" style="position: relative;">p(t)p(t)p(t) is the Dirac comb with period Ts=1fs" role="presentation" style="position: relative;">Ts=1fsTs=1fsT\_s = \frac{1}{f\_s}:xc(t)" role="presentation" style="position: relative;">xc(t)xc(t)x\_c(t) with Fourier transform Xc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), and let the sampled signal be xs(t)=xc(t)p(t)" role="presentation" style="position: relative;">xs(t)=xc(t)p(t)xs(t)=xc(t)p(t)x\_s(t) = x\_c(t)p(t), where p(t)" role="presentation" style="position: relative;">p(t)p(t)p(t) is the Dirac comb with period $T\_s = \frac{xc(t)" role="presentation" style="position: relative;">xc(t)xc(t)x\_c(t) with Fourier transform Xc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), and let the sampled signal bexc(t)" role="presentation" style="position: relative;">xc(t)xc(t)x\_c(t) with Fourier transform $

p(t)=&#x2211;n=&#x2212;&#x221E;&#x221E;&#x03B4;(t&#x2212;nTs)" role="presentation" style="text-align: center; position: relative;">p(t)=∑n=−∞∞δ(t−nTs)p(t)=∑n=−∞∞δ(t−nTs)p(t) = \sum\_{n=-\infty}^{\infty}\delta(t-nT\_s)

The Fourier transform of p(t)" role="presentation" style="position: relative;">p(t)p(t)p(t) is:

P(f)=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;&#x03B4;(f&#x2212;nfs)" role="presentation" style="text-align: center; position: relative;">P(f)=1Ts∑n=−∞∞δ(f−nfs)P(f)=1Ts∑n=−∞∞δ(f−nfs)P(f) = \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty}\delta(f-nf\_s)

The spectrum of the sampled signal is:

Xs(f)=Xc(f)&#x2217;P(f)&#xA0;=&#x222B;&#x2212;&#x221E;&#x221E;Xc(u)P(f&#x2212;u)du&#xA0;=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;Xc(f&#x2212;nfs)" role="presentation" style="text-align: center; position: relative;">Xs(f)=Xc(f)∗P(f) =∫∞−∞Xc(u)P(f−u)du =1Ts∑n=−∞∞Xc(f−nfs)Xs(f)=Xc(f)∗P(f) =∫−∞∞Xc(u)P(f−u)du =1Ts∑n=−∞∞Xc(f−nfs)\begin{align\*}
X\_s(f) &= X\_c(f) \* P(f)\
&= \int\_{-\infty}^{\infty} X\_c(u)P(f-u)du\
&= \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty} X\_c(f-nf\_s)
\end{align\*}

This equation shows that the spectrum of the sampled signal consists of a set of shifted copies of the original spectrum Xc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), spaced apart by the sampling frequency fs" role="presentation" style="position: relative;">fsfsf\_s. Specifically, the n" role="presentation" style="position: relative;">nnnth copy is centered at nfs" role="presentation" style="position: relative;">nfsnfsnf\_s, and the spacing between adjacent copies is fs" role="presentation" style="position: relative;">fsfsf\_s. Mathematically, this can be written as:Xc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), spaced apart by the sampling frequency fs" role="presentation" style="position: relative;">fsfsf\_s. Specifically, the n" role="presentation" style="position: relative;">nnnth copy is centered at nfs" role="presentation" style="position: relative;">nfsnfsnf\_s, and the spacing between adjacent copies isXc(f)" role="presentation" style="position: relative;">Xc(f)Xc(f)X\_c(f), spaced apart by the sampling frequency $

Xs(f)=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;Xc(f&#x2212;nfs)e&#x2212;j2&#x03C0;nfTs" role="presentation" style="text-align: center; position: relative;">Xs(f)=1Ts∑n=−∞∞Xc(f−nfs)e−j2πnfTsXs(f)=1Ts∑n=−∞∞Xc(f−nfs)e−j2πnfTsX\_s(f) = \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty} X\_c(f-nf\_s)e^{-j2\pi nfT\_s}

Note that the spectrum of the sampled signal is periodic with period fs" role="presentation" style="position: relative;">fsfsf\_s, and the spectrum outside of [&#x2212;fm,fm]" role="presentation" style="position: relative;">[−fm,fm][−fm,fm][-f\_m,f\_m] is aliased to the range [&#x2212;fs+fm,fs&#x2212;fm]" role="presentation" style="position: relative;">[−fs+fm,fs−fm][−fs+fm,fs−fm][-f\_s+f\_m,f\_s-f\_m]. To avoid aliasing, the sampling rate must be greater than 2fm" role="presentation" style="position: relative;">2fm2fm2f\_m.fs" role="presentation" style="position: relative;">fsfsf\_s, and the spectrum outside of [&#x2212;fm,fm]" role="presentation" style="position: relative;">[−fm,fm][−fm,fm][-f\_m,f\_m] is aliased to the range [&#x2212;fs+fm,fs&#x2212;fm]" role="presentation" style="position: relative;">[−fs+fm,fs−fm][−fs+fm,fs−fm][-f\_s+f\_m,f\_s-f\_m]. To avoid aliasing, the sampling rate must be greater than $2f\_mfs" role="presentation" style="position: relative;">fsfsf\_s, and the spectrum outside of [&#x2212;fm,fm]" role="presentation" style="position: relative;">[−fm,fm][−fm,fm][-f\_m,f\_m] is aliased to

In summary, the spectrum of the sampled signal is a set of shifted copies of the original spectrum, spaced apart by the sampling frequency, with aliasing occurring if the sampling rate is not greater than twice the maximum frequency of the signal.
한논의 샘플링 정리에 따르면 최대 주파수가 fm" role="presentation" style="위치: 상대적;">fmfmf\_m인 대역 제한 신호는 fs" role="presentation" style="위치: 상대적;">fsfsf\_s, 즉 fs&gt;2fm" role="presentation" style="위치: 상대적;">fs>2fmfs>2fmf\_s > 2f\_m 속도로 취한 샘플로부터 재구성될 수 있다고 명시합니다. 이상적인 샘플링 함수(디락 콤)로 샘플링하는 경우, 샘플링된 신호의 스펙트럼은 다음과 같이 도출할 수 있습니다:fm" role="presentation" style="위치: 상대적;">fmfmf\_m은 fs" role="presentation" style="위치: 상대적;">fsfsf\_s와 같은 속도로 채취한 샘플로부터 재구성될 수 있습니다. 이상적인 샘플링 함수(디락 콤)로 샘플링하는 경우, 스펙트럼 offm" role="presentation" style="위치: 상대적;">fmfmf\_m은 다음에서 가져온 샘플로부터 재구성할 수 있습니다.연속 시간 신호는 푸리에 변환 Xc(f)" role="presentation" style="위치: 상대적;">xc(t)xc(t)x\_c(t)로 하고, 샘플링된 신호는 xs(t)=xc(t)p(t)" role="presentation" style="위치: 상대적;">Xc(f)Xc(f)X\_c(f)로 하십시오: 상대적;">xs(t)=xc(t)p(t)xs(t)=xc(t)p(t)x\_s(t)=x\_c(t)p(t), 여기서 p(t)" role="presentation" style="위치: 상대적;">p(t)p(t)p(t)는 주기 Ts=1fs인 디락 콤(Dirac comb)입니다." role="presentation" style="위치: 상대적;">Ts=1fsTs=1fsT\_s = \frac{1}{f\_s}:xc(t)" role="presentation" style="위치: 상대적;">xc(t)xc(t)x\_c(t), 푸리에 변환 Xc(f)" role="presentation" style="위치: relative;">Xc(f)Xc(f)X\_c(f), 샘플링된 신호는 xs(t)=xc(t)p(t)" role="presentation" style="position: 상대적;">xs(t)=xc(t)p(t)xs(t)=xc(t)p(t)x\_s(t) = x\_c(t)p(t), 여기서 p(t)" role="presentation" style="위치: 상대적;">p(t)p(t)p(t)는 주기 $T\_s = \frac{xc(t)" role="presentation" style="위치: 상대적;">xc(t)xc(t)x\_c(t)는 푸리에 변환 Xc(f)" role="presentation" style="위치: 상대적;">Xc(f)Xc(f)X\_c(f)이고, 샘플링된 신호는 푸리에 변환 $xc(t)" role="presentation" style="위치: 상대적;">xc(t)xc(t)x\_c(t)로 하자.p(t)=&#x2211;n=&#x2212;&#x221E;&#x221E;&#x03B4;(t&#x2212;nTs)" role="presentation" style="text-align: 가운데; 위치: 상대적;">p(t)=∑n=-∞∞δ(t-nTs)p(t)=∑n=-∞∞δ(t-nTs)p(t) = \sum\_{n=-\infty}^{\infty}\delta(t-nT\_s)p(t)의 푸리에 변환"role="presentation" style="위치: 상대적;">p(t)p(t)p(t)p(t)는 다음과 같습니다:P(f)=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;&#x03B4;(f&#x2212;nfs)" role="presentation" style="text-align: center; position: 상대적;">P(f)=1Ts∑n=-∞∞δ(f-nfs)P(f)=1Ts∑n=-∞∞δ(f-nfs)P(f) = \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty}\delta(f-nf\_s)샘플링된 신호의 스펙트럼은 다음과 같습니다:Xs(f)=Xc(f)&#x2217;P(f)&#xA0;=&#x222B;&#x2212;&#x221E;&#x221E;Xc(u)P(f&#x2212;u)du&#xA0;=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;Xc(f&#x2212;nfs)" role="presentation" style="text-align: center; 위치: relative;">Xs(f)=Xc(f)∗P(f) =∫∞-∞Xc(u)P(f-u)du =1Ts∑n=-∞∞Xc(f-nfs)Xs(f)=Xc(f)∗P(f) =∫-∞∞Xc(u)P(f-u)du =1Ts∑n=-∞∞Xc(f-nfs)\begin{align\*}X\_s(f) &= X\_c(f) \* P(f)\.&= \int\_{-\인프티}^{\인프티} X\_c(u)P(f-u)du\.&= \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty} X\_c(f-nf\_s)\end{align\*}이 방정식은 샘플링된 신호의 스펙트럼이 원래 스펙트럼의 시프트 복사본 세트 Xc(f)" role="presentation" style="위치: 상대적;">Xc(f)Xc(f)X\_c(f), 샘플링 주파수 fs" role="presentation" style="위치: 상대적;">fsfsf\_s로 구성됨을 보여줍니다. 구체적으로, n" role="presentation" style="위치: 상대적;">nnn번째 복사본은 nfs" role="presentation" style="위치: 상대적;">nfsnfsnf\_s의 중앙에 있고 인접한 복사본 사이의 간격은 fs" role="presentation" style="위치: 상대적;">fsfsf\_s입니다. 수학적으로는 다음과 같이 쓸 수 있습니다."Xc(f)" role="presentation" style="위치: 상대적;">Xc(f)Xc(f)X\_c(f), 샘플링 주파수 fs" role="presentation" style="위치: 상대적;">fsfsf\_s. 구체적으로, n" role="presentation" style="위치: 상대적;">nnn번째 복사본은 nfs" role="presentation" style="위치: 상대적;">nfsnfsnf\_s의 중심에 있고, 인접한 복사본 사이의 간격은 샘플링 주파수 $\\(f)만큼 간격을 둔 Xc(f)" role="presentation" style="위치: 상대적;">Xc(f)Xc(f)X\_c(f) 입니다.Xs(f)=1Ts&#x2211;n=&#x2212;&#x221E;&#x221E;Xc(f&#x2212;nfs)e&#x2212;j2&#x03C0;nfTs" role="presentation" style="text-align: center; position: 상대적;">Xs(f)=1Ts∑n=-∞∞Xc(f-nfs)e-j2πnfTsXs(f)=1Ts∑n=-∞∞Xc(f-nfs)e-j2πnfTsX\_s(f) = \frac{1}{T\_s}\sum\_{n=-\infty}^{\infty} X\_c(f-nf\_s)e^{-j2\pi nfT\_s}샘플링된 신호의 스펙트럼은 주기 fs" role="presentation" style="위치: 상대;">fsfsf\_s로 주기적이며, [&#x2212;fm,fm]" role="presentation" style="위치: 상대;">[-fm,fm][-fm,fm][-f\_m,f\_m] 범위 밖의 스펙트럼은 [&#x2212;fs+fm] 범위로 앨리어스됩니다.