A notch filter is designed with complex-conjugate zeros placed exactly on the unit circle at frequency ω₀, but no poles are added. What is the main limitation of this design?
AThe filter will be unstable because zeros on the unit circle always cause instability
BThe filter achieves zero gain only at ω₀ but attenuates a very wide band of frequencies around it, not just the target frequency
CThe filter cannot be implemented digitally because zeros on the unit circle require infinite precision
DThe notch depth will be less than 3 dB, making it ineffective for interference rejection
Zeros on the unit circle guarantee zero gain at exactly ω₀ — perfect attenuation at one point. But without poles, the frequency response falls off gradually on both sides, creating a broad notch that attenuates many nearby frequencies you want to preserve. To create a narrow, sharp notch, poles must be added just inside the unit circle at the same angle. The poles partially cancel the effect of the zeros everywhere except at ω₀ itself, dramatically narrowing the affected band.
Question 2 Multiple Choice
In a discrete-time notch filter with poles at radius r near the unit circle, what happens as r approaches 1?
AThe notch becomes broader because the poles move farther from the zeros
BThe filter becomes an all-pass filter because the poles cancel the zeros
CThe notch becomes narrower and sharper (higher Q), but numerical sensitivity increases
DThe filter becomes a resonator, amplifying the target frequency instead of attenuating it
As r → 1, the poles move toward the unit circle, approaching the zeros. Everywhere except at ω₀, the poles increasingly counteract the zeros' attenuation, preserving those frequencies at nearly unit gain. Only at ω₀ do the zeros still dominate. The result is a sharper, higher-Q notch. The cost: when r is very close to 1, tiny coefficient errors (from finite-precision arithmetic) produce large shifts in the effective pole position, dramatically changing notch depth and bandwidth. High-Q designs require careful numerical implementation.
Question 3 True / False
A digital resonator with pole radius r = 1 (poles exactly on the unit circle) produces a stable, sharp-peaked response at the resonant frequency.
TTrue
FFalse
Answer: False
Poles exactly on the unit circle correspond to a marginally stable system — the output grows without bound for an input at the resonant frequency, producing an unstable sinusoidal oscillator, not a stable filter. A practical resonator requires r slightly less than 1, so the poles are just inside the unit circle. This produces a large but finite peak at ω₀ with the peak height and bandwidth controlled by how close r is to 1. As r → 1, the peak grows and bandwidth narrows, but stability is only guaranteed strictly inside the unit circle.
Question 4 True / False
High-Q notch filters require poles placed very close to the unit circle, which introduces numerical sensitivity — small coefficient errors can dramatically degrade notch performance.
TTrue
FFalse
Answer: True
This is an important practical constraint. When r is close to 1, the filter's frequency response is extremely sensitive to the exact pole position. A small change in r or in the cosine coefficient (due to quantization in fixed-point arithmetic) shifts the pole, changing the effective notch frequency, depth, and bandwidth significantly. High-Q designs often require double-precision arithmetic, specialized filter structures (such as coupled-form second-order sections), or careful coefficient wordlength analysis to maintain performance.
Question 5 Short Answer
Explain the relationship between pole radius r and the quality factor Q in a notch filter, and describe the tradeoff that determines how to choose r in practice.
Think about your answer, then reveal below.
Model answer: The pole radius r controls how close the poles are to the zeros on the unit circle. Near the zero at ω₀, both poles and zeros nearly cancel everywhere except at the notch frequency itself. As r increases toward 1, the poles move closer to the unit circle, more aggressively narrowing the affected band — Q rises and bandwidth decreases as approximately Q ≈ ω₀ / (2(1−r)·f_s/2π) for a symmetric notch. The practical tradeoff is between selectivity and numerical robustness: a high-r (high-Q) design isolates a very narrow frequency band but becomes increasingly sensitive to coefficient quantization errors, which can shift the effective notch frequency or reduce attenuation. In practice, r is chosen to achieve the required bandwidth while remaining feasible given the available arithmetic precision.
The geometric picture is key: the notch is narrow because the poles almost cancel the zeros' effect at nearby frequencies. The closer r is to 1, the more complete this near-cancellation is at off-notch frequencies, and the more surgical the attenuation. The numerical sensitivity arises because small perturbations to nearly-canceling pole-zero pairs produce large changes in the residual response — a fundamental sensitivity inherent to high-Q filter structures.