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Digital Signal Processing EC Department, BMCET, Sem 7 Mr. Rahul N. Gonawala

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ASSIGNMENT 1

Submission Date: 22/7/2017

1) State whether the following discrete-time-signals are periodic, and if they are,specify the perioda) x(n) = 2e[ j(n/6-π)]

b) x(n) = cos(n/8) cos(πn/8)c) x(n) = cos(πn/2) - sin(πn/8)d) x(n) = sin2 (40πn)e) x(n) = sin (20πn)+ sin (30πn) + sin (10πn)

2) An analog signal xa(t) = sin(240t) + 3sin(360t) is sampled 600 times persecond.a) Determine the Nyquist sampling rate for xa(t).b) What are the frequencies, in radians, in the resulting discrete time signalx(n)?c) If x(n) is passed through an ideal D/A converter, what is the reconstructedsignal ya(t)?

3) For each of the following signals, determine whether they are(1) static/dynamic;(2) linear/non-linear;(3) time invariant/time varying;(4) causal/non-causal;(5) stable/unstable.

A. y(n) = cos(x(n))

B. ( ) = ∑ ( )4) A discrete –time signal x(n) is given below :

x(n) = { 1, 1, 1, 1, 1, ½ }↑

Sketch and label carefully each of the following signals:(i)x(n-2) (ii) x(4-n) (iii) x(2n) (iv) x(n)u(2-n) (v)x(n-1)δ(n-3)

5) Define ACF (Auto-Correlation Function) of a sequence, explain ACF’s mainProperties and relation with the PSD (Power Spectral Density) of the sequence.


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7) Prove that convergence of absolute sum of the impulse response is a sufficientcondition for BIBO (bounded input bounded output) stability of a LTI system.Show that this condition is also the necessary condition for BIBO stability.

8) X (n) = (-0.5)n

u (n). State whether it is energy or power signal. Justify.


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ASSIGNMENT 2


1) Give the equations for DTFT and inverse DTFT and show that DTFT is alossless (distortion-free) transform.

2) Derive and sketch the DTFT of x[n]= 5cos (0.1πn) + 2cos(0.2πn), over theradian-frequency interval (-3π, 3π).

3) Explain impulse response and step response of a system.Find the step response of a system whose impulse responseis h[n] = a—nu(-n) for 0 < a < 1

4) Find the solution for the given the difference equation y(n) + b2 y(n-2)=0 for n ≥ 0where │b│< 1 and initial conditions are y(-1) = 0, y(-2) = -1

5) Determine the impulse response and the step response of the following causalsystem. Plot the pole-zero patterns and determine if the system is stable.y(n) = (3/4) y(n - 1) - (1/8) y(n - 2) + x(n)

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7) Prove convolution of two sequences in time domain isequivalent to multiplication of two spectra in frequency domain.

8) Realize in Direct Form-II:H(z)=( 1+0.7 z -1 + 0.3 z -2) / ( 1- 0.2 z -1), ROC:0.2< |z|.


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ASSIGNMENT 3


1) Determine the Z-transform of the signals cos ωon u(n) & sin ωon u(n).

2) Determine the difference equation and the impulse response of a system withthe z-transform,H(z)=1/(1-0.5z-1), ROC:0.5< |z|.

3) Determine the inverse-z transform of the function X(z)=log(1+az-1).

4) Given two sequences x1[n]={1, 1}, and x2[n]={1, 2, 1}, find their z-transforms and sketch their corresponding pole-zero plots and try to relatethem.

5) State and prove the differentiation property of Z transform. Using thisproperty , determine Z transform and ROC of signal x(n) = nanu(n)

6) Determine the Circular convolution of the two sequencesx1(n)={2,7,1,8,7} and x2(n)={1,3,2,4,5}

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ASSIGNMENT 4


1) Derive the impulse response of an ideal LPF (low pass filter) with cut-offfrequency of ωc (rad), based on the inverse DTFT.Explain the windowing method for designing an FIR filter.

2) Explain the impulse invariance and bilinear transform based methods fordesigning an IIR filter.

3) Explain the procedure to design a Butter-worth low-pass filter assuming thatsuitable specifications are given.

4) Differentiate between DTFT and DFT.

5) Derive the 8 point Radix 2 DITFFT algorithm to compute DFT. Also explain the bitreversal concept in it.

6) Using DFT and IDFT method compute linear convolution of two sequencesx1(n)={2,3,5 } and x2(n)={1,3 }

7) Explain the window functions used in FIR filter design.

8) Describe the properties of discrete Fourier transform

9) Find the number of complex multiplications and complex additions requiredin the direct computation of 1024 point DFT. How many complexmultiplications and complex additions will be required using FFT algorithm?

10) Find inverse DFT of X(k) = [1 2 3 4 ]

11) List the various methods of designing IIR filter. Explain the bilinear transformationmethod of designing of IIR filter. How does this method overcome the limitation ofother methods?

12) Compare FIR and IIR filters.

13) Derive the equation and discuss 8 point Radix 2 DIT FFT algorithm to compute DFT.Also state the importance of bit reversal table in it.

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ASSIGNMENT 5


1) Explain multiplier-Accumulator (MAC) hardware in DSP processors.

2) Give salient features of DSP architecture.

3) With help of block diagram explain architecture of TMS320C6XXX processor

4) Harvard architecture of DSP processor.

5) Write short note on multiplier-accumulator (MAC) hardware of DSP processor.

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