THEORY EXAMINATION (SEM–VI) 2016-17 DIGITAL SIGNAL PROCESSING

DIGITAL SIGNAL PROCESSING – NEC011

B.Tech (SEM VI) | Section-wise Solved Answers

SECTION – A

(10 × 2 = 20 Marks)

(a) Digital Signal Processing

Digital Signal Processing (DSP) deals with the analysis and processing of signals represented in discrete-time form using digital computation techniques such as algorithms and numerical methods.

(b) Block diagram of DSP

A basic DSP system consists of an analog input signal, followed by sampling and A/D conversion, then digital processing unit, D/A conversion, and finally a reconstruction filter to obtain analog output.

(c) Basic elements required for realization of digital system

The basic elements include adders, multipliers, delay elements (z⁻¹ blocks), and storage registers. These components help implement difference equations.

(d) Linear convolution and its significance

Linear convolution is the process of determining the output of an LTI system by convolving input signal with system impulse response. It represents the physical response of a system to any arbitrary input.

(e) Fundamental time period of

x(t)=sin⁡(15πt)x(t)=\sin(15\pi t)x(t)=sin(15πt)

Angular frequency ω = 15π

T=2π15π=215 secondsT = \frac{2\pi}{15\pi} = \frac{2}{15} \text{ seconds}T=15π2π​=152​ seconds 

(f) Transformation matrix & twiddle factor

A 4×4 DFT matrix consists of powers of twiddle factors.
Twiddle factor:

WN=e−j2πNW_N = e^{-j\frac{2\pi}{N}}WN​=e−jN2π​

Properties include periodicity, symmetry, and conjugate symmetry.

(g) Difference between IIR and FIR filters

IIR filters have feedback and infinite impulse response, while FIR filters have no feedback and finite impulse response. FIR filters are always stable.

(h) Advantages of DSP over ASP

DSP offers high accuracy, flexibility, programmability, better noise immunity, and ease of storage compared to analog signal processing.

(i) Computation efficiency of FFT

FFT reduces computations from N² multiplications (DFT) to N log₂N, improving efficiency significantly.

(j) DFT of sequence

s(n)={1,2,1,3}s(n)=\{1,2,1,3\}s(n)={1,2,1,3}

Using DFT definition,

X(0)=7,X(1)=−j,X(2)=−3,X(3)=jX(0)=7,\quad X(1)=-j,\quad X(2)=-3,\quad X(3)=jX(0)=7,X(1)=−j,X(2)=−3,X(3)=j 

SECTION – B

(Attempt any five – 5 × 10 = 50 Marks)

(a) Parallel form realization

The given transfer function is decomposed using partial fraction expansion. Each term is implemented independently, and outputs are summed to obtain final output.

(b) DFT of x(n)=cos⁡(an)x(n)=\cos(an)x(n)=cos(an)

The DFT is obtained using Euler’s identity. The spectrum shows impulses at ±a depending on periodicity.

(c) DIF FFT flow graph for N=8

In DIF FFT, input is processed directly, and output appears in bit-reversed order. The flow graph has three stages of butterfly computations.

(d) Impulse invariant technique

Given analog system, impulse invariant method samples analog impulse response to obtain digital transfer function. It preserves time-domain characteristics but introduces aliasing.

(e) Butterworth filter design

Order is calculated using magnitude constraints. Using impulse invariant method, poles are mapped into z-plane to obtain H(z).

(f) DIT FFT & computational reduction

Given x(n)=2n,N=8x(n)=2^n, N=8x(n)=2n,N=8.
DIT FFT computes spectrum using recursive decomposition.
Reduction factor:

N2Nlog⁡2N=6424≈2.67\frac{N^2}{N\log_2 N} = \frac{64}{24} \approx 2.67Nlog2​NN2​=2464​≈2.67 

(g) FIR LPF using window

Desired response is truncated and multiplied with given window function to obtain FIR coefficients.

(h) Bilinear transformation

Analog filter is converted using

s=2T1−z−11+z−1s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}s=T2​1+z−11−z−1​

Frequency warping is compensated to obtain required resonant frequency.

SECTION – C

(Attempt any two – 2 × 15 = 30 Marks)

(3)

(i) Ladder structure
Lattice coefficients are computed using recursion, ensuring numerical stability.

(ii) Circular convolution
Using DFT method, circular convolution of

x1(n)={1,2,1,2},x2(n)={3,2,1,4}x_1(n)=\{1,2,1,2\},\quad x_2(n)=\{3,2,1,4\}x1​(n)={1,2,1,2},x2​(n)={3,2,1,4}

is computed efficiently.

(4)

(a) IDFT is used to compute time sequence from given DFT values.
(b)
• Gibbs phenomenon occurs due to truncation of Fourier series.
• Frequency wrapping occurs due to periodicity of discrete spectra.

(5)

(a) Relationship between DFT and Z-transform is derived by sampling Z-transform on unit circle.
(b) Chebyshev filter is designed using bilinear transformation with given ripple constraints.