Multirate DSP Interview Questions

Q1. What is multirate signal processing and why is it used?

Multirate signal processing involves systems where different parts of the signal processing chain operate at different sampling rates, enabling efficient computation and bandwidth management. A digital hearing aid uses decimation to reduce a 48 kHz ADC output to 8 kHz for voice-band processing, saving power and computation. Changing rates allows each processing stage to operate at the minimum rate required, which directly reduces the multiply-accumulate (MAC) operations per second.

Follow-up: What are the two fundamental operations in multirate systems?

Q2. Explain downsampling by a factor M. What happens to the spectrum?

Downsampling by M keeps every Mth sample and discards the rest, compressing the time axis and expanding the frequency axis by M, which causes aliasing if the signal has energy above π/M. Downsampling a 10 kHz signal by 4 on a 40 kHz sampled sequence produces a 10 kHz output with the spectrum stretched to 0–π, and components above 5 kHz alias back. This is why a low-pass anti-aliasing filter with cutoff π/M must precede the downsampler.

Follow-up: What is the cutoff frequency required for the anti-aliasing filter before downsampling by 4 at 48 kHz?

Q3. What is interpolation by a factor L, and what does the spectrum look like after upsampling?

Upsampling by L inserts L-1 zeros between each sample, stretching the time axis and creating L-1 spectral images of the baseband signal in the range 0 to π. After inserting zeros in a speech signal at 8 kHz to produce a 32 kHz sequence (L = 4), the spectrum shows the original signal plus three images at 8, 16, and 24 kHz. A low-pass interpolation filter with cutoff π/L following the upsampler removes these images and scales the signal by L.

Follow-up: Why must the interpolation filter gain be set to L rather than 1?

Q4. What is a polyphase filter and how does it reduce computation in decimation?

A polyphase filter decomposes the anti-aliasing FIR filter h(n) into M sub-filters (polyphase components) Ek(z) = Σ h(Mn+k) z^-n, allowing the filtering to be done at the lower output rate rather than the input rate. A 128-tap FIR decimator by 8 implemented in polyphase form requires only 128/8 = 16 multiplications per output sample compared to 128 at the input rate. The polyphase decomposition exploits the Noble identity to commute the downsampler before the filter sub-blocks.

Follow-up: What is the Noble identity and how is it applied to move the downsampler in a polyphase structure?

Q5. What is the Noble identity in multirate systems?

The Noble identity states that downsampling by M followed by a filter H(z) is equivalent to filtering with H(z^M) followed by downsampling, and dually for upsampling. This identity allows you to move downsamplers before filters and upsamplers after filters in a signal flow graph, enabling polyphase decomposition. In practice, applying the Noble identity to a 256-tap decimation filter allows the entire filtering computation to happen at the low output rate, cutting the MAC count by the decimation factor.

Follow-up: Can the Noble identity be applied to IIR filters in multirate systems?

Q6. Explain the concept of a filter bank and its use in audio coding.

A filter bank splits a wideband signal into multiple sub-band signals using a bank of band-pass filters, allowing each sub-band to be processed or coded independently at a reduced sample rate. The MPEG-1 Layer 3 (MP3) codec uses a 32-band polyphase filter bank to split 44.1 kHz audio before psychoacoustic processing and quantization. Sub-band coding exploits frequency masking properties of human hearing to allocate more bits to perceptually important bands.

Follow-up: What is the difference between a uniform and non-uniform filter bank?

Q7. What is a quadrature mirror filter (QMF) bank?

A QMF bank is a two-channel filter bank where the high-pass filter H1(z) = H0(-z) is the mirror image of the low-pass filter H0(z) around π/2, ensuring the two channels together cover the full spectrum. Johnston QMF coefficients are commonly used in sub-band speech coding, where a 16-tap QMF splits 8 kHz speech into two 4 kHz sub-bands. The QMF condition minimizes aliasing cancellation error, though perfect reconstruction requires additional constraints on the filter.

Follow-up: What condition must the filters satisfy for perfect reconstruction in a QMF bank?

Q8. What is perfect reconstruction in a filter bank?

Perfect reconstruction means that the output of the synthesis filter bank exactly equals a delayed version of the input, with no aliasing, no amplitude distortion, and no phase distortion. In a two-channel PR filter bank, the analysis and synthesis filters must satisfy the matrix condition that the modulation matrix is pseudo-circular. Orthonormal wavelet filter banks, such as those based on Daubechies wavelets, inherently satisfy the perfect reconstruction condition.

Follow-up: Is the Johnston QMF bank a perfect reconstruction system?

Q9. How do you design a sample rate converter for a non-integer ratio, say 44.1 kHz to 48 kHz?

You convert the non-integer ratio 48/44.1 = 160/147 into a rational fraction and implement the sample rate converter as an interpolation by 160 followed by decimation by 147. Direct implementation requires a polyphase filter at 44.1 × 160 = 7.056 MHz, which is impractical, so multistage or farrow-structure interpolation is used instead. The Farrow structure uses polynomial interpolation with continuously variable fractional delay to handle arbitrary ratios efficiently in consumer audio chips.

Follow-up: What is a Farrow structure and what is its advantage for fractional sample rate conversion?

Q10. What is the advantage of multistage decimation over single-stage decimation?

Multistage decimation splits a large decimation factor into smaller factors applied sequentially, reducing the total filter order by allowing each stage to use a relaxed transition band specification. Decimating by 64 in two stages of 8 each requires significantly shorter FIR filters than a single-stage 64× decimator because the transition band relative to each stage's sampling rate is wider. The total computation is lower because most filtering is done at progressively lower rates in later stages.

Follow-up: How do you decide the optimal number of stages and decimation factors at each stage?

Q11. What is a CIC (Cascaded Integrator-Comb) filter and where is it used?

A CIC filter is a multiplier-free decimation or interpolation filter consisting of integrator stages followed by comb stages (or vice versa for interpolation), using only additions and subtractions. The ADXL362 accelerometer and Σ-Δ ADCs like the ADS1256 use CIC filters as the first decimation stage because they require no coefficient storage and can run at very high input rates. The CIC has a sin(x)/x passband droop that must be compensated by a subsequent FIR correction filter.

Follow-up: What is the passband droop characteristic of a CIC filter and how is it compensated?

Q12. Explain the DFT filter bank and its relationship to the STFT.

A DFT filter bank implements N band-pass filters simultaneously using a single N-point DFT, where each DFT bin is the output of a band-pass filter centered at that bin frequency. The Short-Time Fourier Transform (STFT) is a sliding DFT filter bank where the DFT is computed on overlapping windowed frames of the input signal. In real-time audio analyzers and vocoders, the FFT is used to implement the DFT filter bank efficiently, reducing computation from O(N²) to O(N log N).

Follow-up: What is the relationship between the STFT window length and frequency resolution in a DFT filter bank?

Q13. What is oversampling and how does it help in ADC design?

Oversampling samples the analog signal at a rate much higher than the Nyquist rate, spreading quantization noise over a wider bandwidth and reducing the in-band noise power. A Σ-Δ ADC oversampling by a factor of 256 at 12.288 MHz input rate for a 48 kHz audio signal improves the effective resolution by approximately 4 bits (20 log10(√256)/6.02 ≈ 24 dB / 6.02). The oversampled output is then decimated to the Nyquist rate using a digital filter, trading hardware precision for digital processing.

Follow-up: How many bits of resolution does oversampling by 4 add to an ADC?

Q14. What is an analysis filter bank versus a synthesis filter bank?

An analysis filter bank decomposes a wideband input signal into multiple sub-band signals at reduced sample rates, while a synthesis filter bank reconstructs the wideband signal from the processed sub-band signals by upsampling and combining. In an MP3 encoder, the polyphase analysis filter bank splits PCM audio into 32 sub-bands, and in the decoder, the synthesis filter bank reconstructs the audio. For perfect reconstruction, the synthesis filters must be designed in conjunction with the analysis filters to cancel aliasing.

Follow-up: What causes aliasing in a two-channel filter bank and how is it cancelled?

Q15. What is the z-transform of a downsampled signal in terms of the original signal's z-transform?

The z-transform of a signal downsampled by M is Y(z) = (1/M) Σ_{k=0}^{M-1} X(z^(1/M) W_M^(-k)), where W_M = e^(j2π/M) and X(z) is the z-transform of the original signal. This formula shows explicitly that M copies of the spectrum appear in the downsampled output, confirming that aliasing occurs if the signal is not band-limited to π/M before downsampling. This expression is the foundation for deriving polyphase filter structures from first principles.

Follow-up: How does this expression simplify when M = 2 for a two-channel QMF bank analysis?