Quiz: Advanced Filter Design and Implementation
Test your understanding of classical filter approximations, design methods, and multirate filter techniques.
1. What is the defining characteristic of Butterworth filters?
- Equiripple passband response
- Maximally flat passband response with no ripple
- Sharpest possible transition bandwidth
- Constant group delay across all frequencies
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The correct answer is B. Butterworth filters, also known as maximally flat filters, provide the smoothest possible passband response with no ripple and maximum flatness at DC. The magnitude squared frequency response \(|H(\omega)|^2 = 1/(1 + (\omega/\omega_c)^{2N})\) yields a monotonically decreasing response with moderate rolloff. They are excellent general-purpose filters when smooth, predictable frequency response is more important than sharp cutoff transitions.
Concept Tested: Butterworth Filter
2. What tradeoff do Chebyshev Type I filters make compared to Butterworth filters?
- They sacrifice passband flatness (allowing ripple) to achieve steeper rolloff
- They increase filter order to maintain the same rolloff
- They eliminate all stopband attenuation
- They require complex-valued coefficients
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The correct answer is A. Chebyshev Type I filters optimize cutoff sharpness by allowing controlled equiripple in the passband, achieving steeper rolloff than Butterworth filters of the same order. This tradeoff provides sharper transitions at the cost of magnitude response variations (ripple) in the passband, making them suitable when sharp transitions are prioritized over passband flatness, such as channel selection in communication receivers.
Concept Tested: Chebyshev Filter
3. Which classical filter approximation achieves the sharpest possible transition bandwidth for a given filter order?
- Butterworth filter
- Bessel filter
- Chebyshev Type I filter
- Elliptic filter
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The correct answer is D. Elliptic filters (also called Cauer filters) achieve the sharpest possible transition bandwidth for a given filter order by allowing ripple in both passband and stopband. They provide optimal performance when measured by the ratio of transition bandwidth to filter order. This makes them ideal when maximum selectivity is required and both passband and stopband ripple are acceptable, common in biomedical instrumentation and speech processing.
Concept Tested: Elliptic Filter
4. What characteristic do Bessel filters optimize?
- Sharpest magnitude rolloff
- Flattest passband
- Maximally linear phase response (constant group delay)
- Minimum filter order
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The correct answer is C. Bessel filters (also called Thomson filters) optimize phase linearity rather than magnitude response characteristics, providing maximally flat group delay in the passband. This preserves signal waveshapes by maintaining consistent delay across frequencies. The magnitude response is less selective than Butterworth, but the nearly constant group delay minimizes overshoot and ringing, making Bessel filters ideal for pulse transmission, data acquisition, and applications where signal timing is critical.
Concept Tested: Bessel Filter
5. In the FIR window design method, how are filter coefficients computed?
- By solving a system of linear equations
- By truncating and windowing the ideal infinite-length impulse response
- By factoring polynomials in the z-domain
- By random search optimization
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The correct answer is B. The window method designs FIR filters by truncating and windowing the ideal infinite-length impulse response. The procedure: (1) specify desired frequency response \(H_d(\omega)\), (2) compute ideal impulse response via inverse Fourier transform, (3) select window function of length \(M+1\), and (4) compute FIR coefficients \(h[n] = h_d[n]w[n]\). Different windows provide different tradeoffs between transition width and ripple.
Concept Tested: Window Method
See: Window Method
6. What key advantage does the bilinear transform have over impulse invariance for analog-to-digital filter transformation?
- Bilinear transform is faster to compute
- Bilinear transform prevents aliasing through one-to-one frequency mapping, though it introduces frequency warping
- Bilinear transform requires lower filter order
- Bilinear transform only works with FIR filters
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The correct answer is B. The bilinear transform maps the entire s-plane to the z-plane through \(s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}\), providing one-to-one mapping that prevents aliasing (unlike impulse invariance). However, it introduces frequency warping where the analog-digital frequency relationship becomes nonlinear. Pre-warping critical frequencies before analog design compensates for this distortion, making bilinear transform the most widely used analog-to-digital conversion method.
Concept Tested: Bilinear Transform, Impulse Invariance
See: Bilinear Transform
7. What is the primary benefit of polyphase filter implementation for multirate systems?
- Improved frequency response
- Reduced computational complexity by operating primarily at the lower sampling rate
- Elimination of aliasing
- Linear phase guaranteed for all filters
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The correct answer is B. Polyphase decomposition reorganizes filter structures to minimize computation by operating primarily at the lower rate, avoiding operations on discarded (decimation) or zero-valued (interpolation) samples. For decimation by factor M, polyphase reduces multiplications from \(N \cdot f_{in}\) to \(N \cdot f_{out}\), providing computational savings by approximately factor M. This enables efficient implementation of sample rate converters and filter banks.
Concept Tested: Polyphase Filters, Multirate Filters
8. What do filter banks accomplish in signal processing?
- They store multiple filter designs for later use
- They decompose signals into multiple frequency subbands using parallel filter structures
- They combine multiple input signals into one output
- They implement time-domain convolution
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The correct answer is B. Filter banks decompose signals into multiple frequency subbands using parallel filter structures. An M-channel analysis filter bank splits the input into M subbands, each typically downsampled by factor M. Synthesis filter banks reconstruct signals from subband components. Applications include audio compression (MP3, AAC), image compression (JPEG), spectrum analysis, and subband noise reduction where different frequency regions are processed independently.
Concept Tested: Filter Banks
See: Filter Banks
9. Why is the cascade (second-order sections) form preferred over direct form for implementing high-order IIR filters?
- Cascade form requires fewer multiplications
- Cascade form provides excellent numerical properties with low coefficient sensitivity to quantization
- Cascade form only works with even-order filters
- Cascade form eliminates the need for feedback
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The correct answer is B. The cascade form factors the transfer function into second-order sections (biquads): \(H(z) = G\prod_{k=1}^{L} \frac{b_{k0} + b_{k1}z^{-1} + b_{k2}z^{-2}}{1 + a_{k1}z^{-1} + a_{k2}z^{-2}}\). This structure provides excellent numerical properties with low coefficient sensitivity compared to direct form, which exhibits high sensitivity for high-order filters. Cascade form is preferred for practical IIR implementations, especially in fixed-point arithmetic.
Concept Tested: Filter Design Methods
See: Filter Structures
10. What is the main limitation of the impulse invariance method for analog-to-digital filter transformation?
- It cannot transform any analog filters
- It introduces aliasing, limiting it primarily to band-limited filters like low-pass and band-pass
- It only works with Butterworth filters
- It requires complex arithmetic while bilinear transform does not
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The correct answer is B. Impulse invariance creates digital filters by sampling the analog filter's impulse response: \(h[n] = T \cdot h_a(nT)\). While it preserves time-domain characteristics, the frequency response becomes a periodic sum of shifted analog responses, introducing aliasing. This limits impulse invariance primarily to band-limited filters (low-pass and band-pass) when sampling rate significantly exceeds the highest frequency. It cannot design high-pass or band-stop filters without aliasing artifacts.
Concept Tested: Impulse Invariance