downsampling
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downsampling refers to the process of decreasing the effective sampling rate the doing that in the discrete-time system so its after weve initially converted the signal from continuous time to discrete time were going to reduce the effective sampling rate of the signal to set this up were going to look at first what happens if we sample a signal at intervals of tea or a sampling frequency of two PI over T radians per second then were going to look at how the spectrum compares when we sample the same signal at intervals of M times T or a lower sampling rate by a factor of M so here I have a signal with a spectrum X of Omega and it basically extends from minus Omega naught to maintenant has unit amplitude here and if we apply the sampling theorem to this signal we find that the Fourier transform of the sampled signal excess of Omega has these replicates that occur at multiples of Omega s the sampling frequency so theres many more Im just not showing them all here well when we convert this to discrete time frequency lower case Omega using the relationship that lower case Omega is upper case Omega times T then I obtain the discrete-time fourier transform of our sampled signal X via the J Omega the signal repeats now at intervals of two pi then the edge of the original signal spectrum at Omega naught gets mapped into 2 pi times Omega naught divided by Omega sub S or Omega naught
times T so thats what happens when we sample a signal intervals of T now if instead we sample that same signal at intervals of M times capital T where M is an integer we call that X tilde of n we can write the Fourier transform of the sampled signal X tilde sub s of Omega as given here and instead of repeating at Omega sub s this repeats at the new sampling frequency Omega sub s tilde which is Omega sub s divided by M so we have replicates occurring at multiples of Omega sub s divided by M and because M is an integer these replicates occur closer to the bandwidth of the original signal which was a mega nought and then the amplitude of the signal gets scaled by a factor of M because the sampling rate is M times T we can convert this to discrete time frequency lower case Omega by again using the transformation that lower case Omega is equal to upper case Omega times the sampling interval but this time the sampling of interval is M times T we obtain the dtft X tilde v2j Omega which is again 2 pi periodic so what happened the replicates at the sampling frequency show up at 2 pi and 4 pi and so on but now this band edge is going to be M times T Omega naught and so effectively this signal spans a bandwidth M times as wide as it did over here when we sample that intervals of T
sampling intervals of M times T causes this to stretch out by a factor of M and the amplitude decreases by a factor of M as well so this is what we seek to accomplish when we down sample the signal we basically want to go from a discrete-time fourier transform which has this particular shape here and convert it to a discrete time Fourier transform on like we have on the right where things are scaled on the frequency axis by a factor 2 M within the minus PI to PI bandwidth and the amplitude is also scaled by a factor of M a system that will implement down sampling is shown here were going to take our signal X of n which is already been sampled say at a rate T and were going to first pass it through a low-pass filter and thats going to prevent aliasing of components that were fine and X of n but at the lower sampling rate would alias then we have this signal W of n thats been low-pass filtered and were going to throw away every value except the multiples of so were going to discard all the values that have and not equal to M times L and that gives us a signal Y of n which is just W of M times n so this is like a scaling of the time axis by a factor of M in the frequency domain we have X of e to J Omega will show that it has amplitude
1 over T and the part of the signal were interested in goes from minus PI over m 2 PI over m well assume that theres other frequency components out here in this pi over m 2 pi interval and if we were just to change the sampling rate without doing any filtering these components would alias back into the band of interest so first were going to low-pass filter with a pass band going from minus PI over m to pi over m and that will get rid of these components at higher frequencies out here that could a Leas so now we have a clean signal with essentially no frequency components in the PI over m 2 pi interval and were going to pass this signal W of n through a operation that we throw away all the values except them integer multiples of M and that produces Y of either J Omega what happens in the frequency domain when we throw away all the values except the ones that are multiples of M is we stretch this signal out weve compressed it in time were going to stretch it in frequency and now instead of ending at PI over m for the band edge its going to extend all the way out to pi so the property when you throw away samples is written down here at the bottom of the page if Y of n is W of M times n then the discrete-time fourier transform Y IV the J Omega is 1 over
m times the sum of lowercase M equals 0 to uppercase M minus 1 of W of either the J quantity Omega minus M 2 pi divided by Capital m so this division by M is what introduces the scaling of the frequency axis stretching things by a factor we look at these sequence of operations in the time domain you see its really quite straightforward we take our signal X of N and we low-pass filter it which is going to smooth the signal out and get rid of some of the higher frequency fluctuations that we see and then were going to throw away all the values except the multiples of capital m and in this case Im illustrating M equals 3 so were going to keep this value at 0 the next value were going to keep was located at 3 and W but becomes 1 in Y and then at 6 in W becomes 2 and y 9 and W becomes time 3 and y 4 notation we sometimes depict this block where we discard sync every M sample by the symbol a down arrow with a factor of M and that just indicates this process of throwing away samples in between those that are integer multiples of M so this is quite simple to do in a discrete-time system and this provides us flexibility in the discrete-time system for changing the sampling rate to accomplish different goals such as reducing the strictness of the specifications on the analog low-pass filter thats used for anti-aliasing
tags:
downsampling, decimation, multirate signal processing
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