Sampling of Discrete Time Signals

Sampling of discrete-time signals takes every M-th sample from an existing discrete sequence to produce a lower-rate sequence, a process called downsampling or decimation. This operation appears in multirate signal processing systems used in audio codecs, wireless receivers, and image compression.

Core Concept

Discrete-time sampling, also called downsampling or decimation, selects one sample out of every M samples from an input sequence x[n] to form an output sequence y[m] = x[Mn]. The output sequence has a lower sample rate by a factor M. This is a purely discrete operation requiring no analog hardware.

In the frequency domain, downsampling by M stretches the DTFT spectrum by M and adds M-1 spectral replicas. If the original sequence has spectral content above pi/M radians, those replicas overlap the baseband spectrum. This is the discrete-time version of aliasing. A digital anti-aliasing filter with cutoff pi/M must precede the downsampler to prevent this.

Upsampling inserts M-1 zeros between each sample to increase the sample rate by M. The upsampled sequence has spectral images at multiples of 2*pi/M. An interpolation filter removes those images and smooths the upsampled sequence. Together, a downsampler and an upsampler form the building blocks of every multirate system.

Key Formula

The DTFT relationship for downsampling by integer M is:

Y(omega) = (1/M) * sum_{k=0}^{M-1} X((omega - 2*pi*k) / M)

Here X(omega) is the DTFT of x[n] and Y(omega) is the DTFT of y[m] = x[Mn]. The M terms in the sum are the M spectral replicas at intervals of 2*pi/M in the stretched spectrum. Aliasing-free decimation requires X(omega) = 0 for |omega| > pi/M.

Problem: x[n] = {4, 3, 2, 1, 5, 6, 2, 4} (n = 0 to 7). Downsample by M = 2.

Formula: y[m] = x[2m]

Step 1: List input samples with indices
  x[0]=4, x[1]=3, x[2]=2, x[3]=1, x[4]=5, x[5]=6, x[6]=2, x[7]=4

Step 2: Keep every 2nd sample (even indices)
  m=0: y[0] = x[0] = 4
  m=1: y[1] = x[2] = 2
  m=2: y[2] = x[4] = 5
  m=3: y[3] = x[6] = 2

Step 3: Output sequence
  y[m] = {4, 2, 5, 2}

Step 4: Sample rate comparison
  If input rate = Fs samples/sec, output rate = Fs/2 samples/sec

Final Answer: y[m] = {4, 2, 5, 2}, length reduced from 8 to 4.

Exam Tip: The DTFT of the downsampled sequence contains M stretched-and-summed copies of the original spectrum. To avoid aliasing in discrete-time downsampling, the digital pre-filter must cut off at omega = pi/M, not at pi. Many GATE problems ask whether aliasing occurs for a given input spectrum and downsampling factor; check if X(omega) is non-zero for |omega| > pi/M.

Properties Summary

• Downsampling definition: y[m] = x[Mn], keeps one in every M samples.
• DTFT of downsampled signal: Y(omega) = (1/M)*sum_{k=0}^{M-1} X((omega - 2*pi*k)/M).
• Aliasing condition: occurs when X(omega) is non-zero for |omega| > pi/M.
• Pre-filter requirement: digital LPF with cutoff pi/M before the M:1 downsampler.
• Upsampling: y[m] = x[m/L] for m a multiple of L, zero otherwise; inserts L-1 zeros.
• Upsampling DTFT: Y(omega) = X(L*omega), spectral compression with images at 2*pi/L intervals.
• Noble identity: a digital filter H(z^M) followed by a downsampler M equals a downsampler M followed by H(z). This allows efficient polyphase implementation.

Quick Revision

• Decimation by M: y[m] = x[Mn], output rate = input rate / M.
• Aliasing-free condition: signal bandwidth must be below pi/M radians.
• Upsampling by L: insert L-1 zeros between samples, then apply interpolation LPF.
• Multirate system: combines upsampler and downsampler for sample rate conversion.
• Noble identity enables polyphase decomposition for computationally efficient filters.
• In DTFT, downsampling stretches the spectrum; upsampling compresses it.
• Exam trap: confusing the discrete-time anti-aliasing filter cutoff (pi/M radians) with the continuous-time cutoff (fs/2 Hz). They are different quantities in different domains.