# Understanding index transformation in derivation of Fourier transform for sampling rate reduction

Was going over some notes regarding deriving fourier transform equation for Sampling Rate Reduction. Reference to Notes from below link https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-341-discrete-time-signal-processing-fall-2005/lecture-notes/lec05.pdf or from Book Discrete-Time Signal Processing by Alan V. Oppenheim (2nd Edition), equation 4.75.

$$r = i + kM$$

I am lost as to how this is obtained. I understand that every $$M-1$$ samples are dropped from original sampling results; but still cannot understand how this expression for $$r$$ is derived.

Could someone help me understand this?

It's not derived, it's just chosen in a smart way such that the relationship between the decimated and the original sequences becomes obvious.

It's just a rearrangement of the terms of the sum. As a simple example, take an infinite sum of numbers $$a_r$$:

$$S=\sum_{r=-\infty}^{\infty}a_r\tag{1}$$

Under certain conditions that we don't need to bother with now we can rearrange the sum and write it like

\begin{align}S&=\ldots+a_{-2M}+a_{-M}+a_0+a_M+a_{2M}+\ldots\\&\ldots+a_{-2M+1}+a_{-M+1}+a_1+a_{M+1}+a_{2M+1}+\ldots\\&\vdots\\&\ldots+a_{-2M+(M-1)}+a_{-M+(M-1)}+a_{0+(M-1)}+a_{M+(M-1)}+a_{2M+(M-1)}+\ldots\tag{2}\end{align}

with some arbitrarily chosen integer $$M$$. So we just start with element $$a_0$$ and add every $$M^{th}$$ element, then we move to element $$a_1$$ and add again every $$M^{th}$$ element, etc. If we do this $$M$$ times, we've added all elements, just like in the originial sum $$(1)$$.

Eq. $$(2)$$ can be written as

$$S=\sum_{i=0}^{M-1}\sum_{k=-\infty}^{\infty}a_{i+kM}\tag{3}$$

which means that we expressed the index $$r$$ as $$r=i+kM$$.

• thanks for info. What troubles me is that the down sampling samples only contain a(M) + a(2M) + a(3M) ... so in context of summation i would write it as $S=\sum_{k=-\infty}^{\infty}a_{kM}\tag{1}$ Jun 17 '20 at 18:54
• @niil87: But that sum is not over time domain samples, this is a sum of shifted frequency spectra, there's no downsampling involved in that sum. Jun 17 '20 at 19:14
• thank you for reply, makes sense! Jun 19 '20 at 9:02