Questions: Interpolation, Image Rejection, and Upsampling

Question 1

A discrete signal originally sampled at 10 kHz is upsampled by factor L = 4 via zero-insertion (before any filtering). Where do the spectral images appear in the output spectrum?

• A. At the new Nyquist frequency of 20 kHz only
• B. At multiples of the original sampling rate — near 10 kHz, 20 kHz, and 30 kHz — within the new 0–20 kHz range
• C. Images do not appear — zero-insertion only increases the sample rate without altering the spectrum
• D. At multiples of the new sampling rate, 40 kHz apart

Answer: (B)

Upsampling by L=4 produces an output at 4×10 kHz = 40 kHz. The DTFT of the upsampled sequence is periodic, and zero-insertion causes the original baseband spectrum (0–5 kHz) to repeat at multiples of the original sampling frequency (10 kHz) within the new frequency range. Images appear near 10 kHz, 20 kHz, and 30 kHz — three unwanted copies of the baseband. The anti-imaging filter with cutoff at the original Nyquist (5 kHz) removes these images while preserving the baseband.

Question 2

A DSP engineer upsamples a 44.1 kHz audio signal by factor 4 to produce a 176.4 kHz stream but forgets to apply the anti-imaging filter. What is the result?

• A. Exactly the same audio — the filter is only needed for final analog reconstruction
• B. Improved audio quality because the higher sample rate adds resolution between original samples
• C. The original audio plus distortion from spectral images appearing as spurious high-frequency content aliased into the signal
• D. Silence — the upsampled signal has no energy at the original frequencies

Answer: (C)

Without the anti-imaging filter, the spectral images created by zero-insertion remain in the signal. These images are copies of the original audio spectrum appearing at multiples of 44.1 kHz within the new frequency range. When the signal is subsequently processed or reconstructed, these images cause audible distortion. The upsampling itself adds no new information — zero-inserted samples contribute nothing real. The anti-imaging filter is mandatory to produce a valid higher-rate signal.

Question 3

Inserting zeros between samples during upsampling creates new signal information, which is why higher sample rates can represent audio with greater accuracy.

• T. True
• F. False

Answer: False

Zero insertion adds no information — the inserted zeros are mathematically inert placeholders. What upsampling creates is spectral images (artifacts), not new signal content. The anti-imaging filter then produces interpolated values between the original samples, but these are derived from the existing signal data — they represent the best estimate of what the original continuous signal was doing between samples, not new information. A student who confuses zero-insertion with information gain will misunderstand what interpolation actually achieves.

Question 4

The anti-imaging filter applied after upsampling by L should have its cutoff at the original Nyquist frequency (f_s/2), not at the new Nyquist frequency (L·f_s/2).

• T. True
• F. False

Answer: True

The goal of the anti-imaging filter is to retain only the original baseband spectrum (0 to f_s/2) and suppress all spectral images at higher frequencies. If the cutoff were set at the new Nyquist (L·f_s/2), the filter would pass all the images it is supposed to remove. The original Nyquist frequency is the boundary between the true signal and the artifacts, so that is the correct cutoff. The filter gain must also be L to compensate for the energy reduction caused by inserting zero-valued samples.

Question 5

Explain why upsampling by inserting zeros creates spectral images, and what role the anti-imaging filter plays in producing a valid higher-rate signal.

Model answer: The DTFT of any discrete sequence is periodic with period equal to the sampling rate. When L−1 zeros are inserted between each sample, the resulting sequence has the same baseband spectrum but now that spectral period structure repeats L−1 additional times within the wider frequency range of the higher-rate output. These repetitions — spectral images — are mathematical artifacts of the zero-insertion operation, not real signal components. The anti-imaging (lowpass) filter with cutoff at the original Nyquist frequency passes only the true baseband copy and suppresses all images. After filtering, the output contains interpolated values between the original samples (the filter effectively fills in values consistent with a bandlimited reconstruction of the original signal), producing a legitimate higher-rate representation without any spurious content.

This is the discrete-time analog of the continuous-time reconstruction process: just as a DAC reconstruction filter suppresses aliased copies in the continuous domain, the anti-imaging filter suppresses copies in the discrete domain after upsampling. Both are mandatory; both remove the same type of artifact from the same spectral periodicity.