SECTION A – Very Short Questions (2×10 = 20)
(a) Distinguish between IIR and FIR filter.
IIR (Infinite Impulse Response) Impulse response is infinite in duration (uses feedback).
Has both poles and zeros. Generally lower order for same specs → more efficient.
May be unstable if poles are outside unit circle.
Linear phase is difficult to obtain. FIR (Finite Impulse Response)
Impulse response is finite (no feedback, only feed-forward).
Has only zeros (no poles except at origin). Always stable.
Easy to design exact linear phase. Usually higher order for same specs → more multiplications.
(b) Define linear phase response of a filter.
A discrete-time filter has linear phase if its phase response is of the form
ϕ(ω)=−ωn0+ϕ0\phi(\omega) = -\omega n_0 + \phi_0ϕ(ω)=−ωn0+ϕ0
i.e. phase is a straight line versus frequency.
This means all frequency components are delayed by the same constant group delay ⇒ no phase distortion in the signal.
(c) Compare bilinear transformation and impulse invariant method of IIR filter design.
Bilinear Transform (BLT)
Maps analog sss-plane to digital zzz-plane by:
- s=2T 1−z−11+z−1s = \frac{2}{T}\,\frac{1 - z^{-1}}{1 + z^{-1}}s=T21+z−11−z−1
No aliasing (one-to-one mapping of jΩ to e^{jω}). Frequency axis is warped → need pre-warping.
Always maps LHP (stable analog) to inside unit circle (stable digital).
Impulse Invariance
Matches time-domain impulse response samples: hd[n]=ha(nT)h_d[n] = h_a(nT)hd[n]=ha(nT).
Simple for low-pass / band-limited analog filters. Suffers from aliasing in frequency domain.
No frequency warping (straight mapping, but many analog bands fold).
(d) Distinguish between Butterworth and Chebyshev (Type-I) filter.
Butterworth Filter
Magnitude response is maximally flat in passband (no ripple). Smooth monotonic roll-off.
Slower transition (needs higher order for sharp cut-off). Equal ripple = 0 in passband & stopband.
Chebyshev Type-I Filter
Has equiripple passband and monotonic stopband.
Sharper transition than Butterworth for same order. Some passband ripple (controlled by ε).
Group delay and phase are more distorted.
(e) Need for employing window for designing FIR filter.
Ideal frequency responses (e.g. ideal LPF) have infinite-length impulse responses.
To implement in practice, we must truncate to finite length.
Direct, sharp truncation causes Gibbs phenomenon (ripples).
A window function (Hamming, Hanning, Blackman, Kaiser etc.) smoothly tapers the impulse response to zero, which
Controls side-lobe levels (ripple) Controls main-lobe width (transition width)
Gives a trade-off between ripple and bandwidth.
(f) Why simple truncation of the ideal IIR impulse response is usually undesirable?
Abrupt truncation = multiply by a rectangular window.
In frequency domain this becomes convolution with sinc, causing:
Large ripples in passband & stopband (strong Gibbs phenomenon).
Poor control over side-lobes and stopband attenuation.
May distort the desired response and fail specifications.
Better approach: use smoother windows (Hamming, Kaiser…) or optimal methods (e.g. Parks–McClellan).
(g) Develop the 4-point DFT of the sequence x(n)={1,1}x(n) = \{1,1\}x(n)={1,1}.
Assume 4-point sequence:
x[0]=1, x[1]=1, x[2]=0, x[3]=0x[0]=1,\ x[1]=1,\ x[2]=0,\ x[3]=0x[0]=1, x[1]=1, x[2]=0, x[3]=0
4-point DFT:
X[k]=∑n=03x[n]e−j2πkn/4,k=0,1,2,3X[k] = \sum_{n=0}^{3} x[n] e^{-j 2\pi kn/4},\quad k=0,1,2,3X[k]=n=0∑3x[n]e−j2πkn/4,k=0,1,2,3
X[0]=1+1+0+0=2X[0] = 1 + 1 + 0 + 0 = 2X[0]=1+1+0+0=2
X[1]=1+1e−jπ/2=1−jX[1] = 1 + 1 e^{-j\pi/2} = 1 - jX[1]=1+1e−jπ/2=1−j
X[2]=1+1e−jπ=1−1=0X[2] = 1 + 1 e^{-j\pi} = 1 - 1 = 0X[2]=1+1e−jπ=1−1=0
X[3]=1+1e−j3π/2=1+jX[3] = 1 + 1 e^{-j3\pi/2} = 1 + jX[3]=1+1e−j3π/2=1+j
So
X[k]={ 2, 1−j, 0, 1+j }X[k] = \{\,2,\ 1-j,\ 0,\ 1+j\,\}X[k]={2, 1−j, 0, 1+j}
(h) How is FFT algorithm applied to compute the inverse DFT?
To compute IDFT using FFT:
Use the property:
- x[n]=1NIDFT{X[k]}=1N(DFT{X∗[k]})∗x[n] = \frac{1}{N} \text{IDFT}\{X[k]\} = \frac{1}{N}\Big(\text{DFT}\{X^{*}[k]\}\Big)^{*}x[n]=N1IDFT{X[k]}=N1(DFT{X∗[k]})∗
where * means complex conjugate.
Steps:
Take conjugate of the DFT sequence X[k]X[k]X[k]. Compute N-point FFT of this conjugated sequence.
Take conjugate of the FFT output. Divide each sample by N → gives the IDFT result.
So we reuse the same FFT hardware/software for inverse transform.
(i) Highlight the features of a commercial digital signal processor (DSP).
Common DSP chip features:
Harvard architecture (separate program & data memories).
MAC unit (single-cycle Multiply–Accumulate).
Hardware support for circular buffers (for FIR, IIR).
Zero-overhead looping (no extra cycles for loop control).
Special addressing modes: bit-reversed, modulo, auto-increment/decrement.
On-chip peripherals: ADC/DAC interfaces, serial ports (I²S, SPI), DMA.
Often fixed-point or floating-point arithmetic optimized for real-time signal processing.
(j) Explain the concept of multistage sampling rate conversion.
Multistage sampling rate conversion = performing large sample-rate change as a cascade of several small-factor stages. Example: instead of decimating by 12 directly, do
decimate by 3, then by 4 (or 2×2×3), with suitable anti-alias filters at each stage.
Benefits:
Lower filter order per stage → fewer multiplications.
Overall lower computational complexity than single large-factor converter.
Easier filter design and better control of passband/stopband specs.
Used in sample-rate converters, multirate filter banks, and modern communication systems.
Quick Guidance for Long Questions (Sections B–C)
I won’t fully solve all numericals here, but this is how you should approach them:
Realization questions (cascade/parallel, direct form I/II, ladder)
→ First express H(z)H(z)H(z) as ratio of polynomials in z−1z^{-1}z−1, factor into first/second-order sections, then draw block diagrams.
Analog → digital IIR design (bilinear / impulse invariance)
→ Start from Ha(s)H_a(s)Ha(s), apply the mapping (BLT or impulse invariance), simplify to get H(z)H(z)H(z), then express in powers of z−1z^{-1}z−1.
FIR design with windows (rectangular, Hamming, Kaiser)
→ Determine ideal impulse response hd[n]h_d[n]hd[n], apply window w[n]w[n]w[n], then shift to make it causal.
DFT / FFT questions
→ For given time sequence, split into even/odd parts (DIT) or first/second half (DIF), apply butterfly structure.
Overlap-add method
→ Break long input into blocks, convolve each block (via DFT) with h(n)h(n)h(n), then shift and add overlaps.
Multirate & adaptive algorithms (RLS, LMS)
→ Focus on block diagrams, basic equations, advantages, and applications.
