FUNDAMENTALS OF DIGITAL SIGNAL PROCESSING (EEN011)
SECTION – A
(Short Answer Questions | 2 Marks Each)
(a) Properties of ROC
The Region of Convergence (ROC) of a Z-transform is a ring or region in the z-plane where the transform converges. It does not include poles, is connected, and determines system stability and causality.
(b) Energy or Power Signal
Given: x(t) = A for 0 < t < T₀, and 0 otherwise
Energy = ∫|x(t)|²dt = A²T₀ (finite) Power = 0
Hence, x(t) is an energy signal.
(c) Even and Odd Components of x(n) = e⁻ⁿ⁄⁴u(n)
Even part: xₑ(n) = ½[x(n) + x(−n)]
Odd part: xₒ(n) = ½[x(n) − x(−n)]
The signal is neither purely even nor odd.
(d) Continuous-Time Signal & Product of Odd Signals
A continuous-time signal is defined for all real values of time.
If x₁(t) and x₂(t) are odd signals, then: x₁(−t)x₂(−t) = x₁(t)x₂(t)
Hence, product of two odd signals is an even signal.
(e) Zero Padding
Zero padding is the process of appending zeros to a signal sequence.
It improves frequency resolution, simplifies FFT computation, and helps in linear convolution using DFT.
(f) Twiddle Factor
Twiddle factor is defined as: Wₙ = e⁻ʲ²π⁄ᴺ
For N = 4: W₄ = e⁻ʲπ⁄² = −j
(g) Frequency Warping & Bilinear Transformation
Frequency warping occurs due to nonlinear mapping of analog frequency to digital frequency.
It is overcome by pre-warping the analog frequencies before applying bilinear transformation.
(h) Sampling Theorem
A continuous-time signal can be perfectly reconstructed if sampling frequency Fs ≥ 2Fm, where Fm is the maximum frequency of the signal.
(i) Band-Pass and Band-Stop Filter
A band-pass filter allows a certain frequency band and attenuates others.
A band-stop filter rejects a specific frequency band while passing low and high frequencies.
(j) Input Quantization Error
Quantization error is the difference between actual input value and its quantized value. It introduces noise in digital systems.
SECTION – B
(Descriptive Questions | 10 Marks Each)
(a) DTFT Properties Without Computation
Given x(n) = (1,2,3,2,1):
(i) X(eʲ⁰) = sum of x(n) = 9 (ii) ∠X(eʲω) = 0 (signal is symmetric and real)
(iii) X(eʲπ) = 1 − 2 + 3 − 2 + 1 = 1 (iv) ∫X(eʲω)dω = 2πx(0)
(v) ∫|X(eʲω)|²dω = 2π∑|x(n)|²
(b) LTI System Stability
Given:
y(n) = 0.6y(n−1) − 0.08y(n−2) + x(n) (i) h(n) is obtained by solving characteristic equation.
(ii) Since poles lie inside unit circle, the system is BIBO stable.
(c) DTFT of x(n) = a|n| , 0 < a < 1
DTFT exists since signal is absolutely summable. Magnitude spectrum decays smoothly.
Fourier Transform of signum function: FT{sgn(t)} = 2⁄(jω)
(d) Graphical Convolution
Given: h(t) = u(t), x(t) = e⁻ᵃᵗu(t)
Output: y(t) = ∫₀ᵗ e⁻ᵃτ dτ = (1 − e⁻ᵃᵗ)/a · u(t)
(e) DFT Using FFT Algorithms
The sequence x(n) = {1,2,3,4,4,3,2,1} is decomposed using:
DIT FFT (time decimation) DIF FFT (frequency decimation)
Both yield the same DFT, verifying correctness.
(f) Autocorrelation Estimation
Autocorrelation sequences are estimated using: Biased estimator
Unbiased estimator
These are used for power spectral density estimation of random signals.
(g) Impulse Invariance Technique
Impulse invariance maps analog impulse response into digital domain.
Disadvantage: aliasing of frequency response.
This can be reduced using pre-filtering.
(h) Overlap-Add and Overlap-Save
These methods perform linear convolution on long data sequences using FFT.
Overlap-add overlaps output blocks, while overlap-save discards corrupted samples.
SECTION – C
(Long Answer Questions | 15 Marks Each)
3. Short Notes
(i) Oversampling & Noise Shaping
Oversampling spreads quantization noise over a wider bandwidth. Noise shaping pushes noise out of the signal band, improving resolution.
(ii) Coefficient Quantization Effect
Quantization of filter coefficients introduces amplitude and phase distortion and may cause instability.
(iii) Discrete Cosine Transform (DCT)
DCT converts signals into sum of cosine functions and is widely used in image and video compression due to energy compaction.
4.
(i) Frequency Response of Rational System
Frequency response is obtained by evaluating system function H(z) on unit circle. Poles and zeros determine magnitude and phase response.
(ii) FIR Filter Design Using Hamming Window
Given:
Fc = 500 Hz, Fs = 2000 Hz
Normalized frequency = 0.25
Ideal impulse response is multiplied by Hamming window to obtain FIR coefficients.
5. Sampling of Analog Signal
Given xₐ(t) = 10 sin(100πt + π/3)
(i) Minimum Fs = 100 Hz
(ii) For Fs = 200 Hz, discrete-time frequency ω = π/2
(iii) For Fs = 75 Hz, aliasing occurs
(iv) Aliased frequency gives same samples as part (iii)
