STUPIDSID
1 (a)
Draw the block diagram of a typical Digital Signal Processing system and explain.
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1 (b)
A discrete-time signal x(n) is given below :
\[X (n) = \{1, 1,\underset {\uparrow }{1}, 1, 1,1/2\}\]
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)
\[X (n) = \{1, 1,\underset {\uparrow }{1}, 1, 1,1/2\}\]
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)
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2 (a)
Perform the liner convolution of the following sequences:\[X_{1}(n) = \{1, \underset {\uparrow }{2}, 3, 4,5\},\ \ X_{2}(n) = \{-1,0, \underset {\uparrow }{1}\}\]
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2 (b)
(i) For the following system, determine whether the system is stable,causal,linear,time invariant,memoryless:
\[T\left \{ x(n) \right \}=\sum_{k=n_{0}}^{n}x(k)\]
What are the advantages of digital signal processing over analog signal processing>
\[T\left \{ x(n) \right \}=\sum_{k=n_{0}}^{n}x(k)\]
What are the advantages of digital signal processing over analog signal processing>
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2 (c)
Let X(ejw ) denote the fourier transform of the signal x(n) .Perform the following calculations without explicitly evaluating X(ejw )
\[X_{1}(n) = \{-1,0,1, \underset {\uparrow }{2}, 1, 2,1,0,-1\}\]
i) Evaluate X(ejw ) ?w=0
ii) Evaluate X(ejw ) ?w=π
iii) Find θ X((ejw))
iv) Evaluate \[\int_{-\pi}^{\pi}\limits X(e^{jw})dw\]
v) Determine and sketch the signal whose fourier transform is X(e- jw )
vi) Determine and sketch the signal whose fourier transform is Re{X(ejw )
\[X_{1}(n) = \{-1,0,1, \underset {\uparrow }{2}, 1, 2,1,0,-1\}\]
i) Evaluate X(ejw ) ?w=0
ii) Evaluate X(ejw ) ?w=π
iii) Find θ X((ejw))
iv) Evaluate \[\int_{-\pi}^{\pi}\limits X(e^{jw})dw\]
v) Determine and sketch the signal whose fourier transform is X(e- jw )
vi) Determine and sketch the signal whose fourier transform is Re{X(ejw )
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3 (a)
Determine the z-transform of the following sequences. Sketch ROC and pole zero plot :
(i)x1 (n) = α|n|, 0 < ?α ? < 1
(ii)x2 (n) = (-1/3)n u(n) - (1/2)n u(-n-1)
(i)x1 (n) = α|n|, 0 < ?α ? < 1
(ii)x2 (n) = (-1/3)n u(n) - (1/2)n u(-n-1)
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3 (b)
Suppose the z-transform of x(n) is
\[X(z)=\dfrac{z^{10}}{(z-(1/2))(z-(3/2))^{10}(z+(3/2))^{2}(z+(5/2))(z+(7/2))}\]
It is also known that x(n) is a stable sequence.
(i)Determine the region of convergence of X(z).
(ii) Determine x(n) at n = -8.
\[X(z)=\dfrac{z^{10}}{(z-(1/2))(z-(3/2))^{10}(z+(3/2))^{2}(z+(5/2))(z+(7/2))}\]
It is also known that x(n) is a stable sequence.
(i)Determine the region of convergence of X(z).
(ii) Determine x(n) at n = -8.
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3 (c)
Consider the discrete time system with an ideal low pass filter with cutoff frequency π/8 radian/s.
IMAGE
(i)If xc (t) is bandlimited to 5 kHz , what is the maximum value of T that will avoid aliasing?
(ii)If 1/T = 10 kHz , what will the cutoff frequency of the continuous-time filter be?
(iii) Repeat part
(iv) for 1/T=20kHz.
IMAGE
(i)If xc (t) is bandlimited to 5 kHz , what is the maximum value of T that will avoid aliasing?
(ii)If 1/T = 10 kHz , what will the cutoff frequency of the continuous-time filter be?
(iii) Repeat part
(iv) for 1/T=20kHz.
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3 (d)
Draw the structures of the following discrete time system:
\[H(z)=\dfrac{(1+z^{-1})^{2}}{1-0.75z^{-1}+0.125z^{2}}\]
(i)Direct form - I
(ii)Direct Form - II
(iii)Cascade form
(iv)Parallel form.
\[H(z)=\dfrac{(1+z^{-1})^{2}}{1-0.75z^{-1}+0.125z^{2}}\]
(i)Direct form - I
(ii)Direct Form - II
(iii)Cascade form
(iv)Parallel form.
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4 (a)
Discuss the following transformation methods to design digital filters:
(i)Impulse invariance (ii)Bilinear transformation
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4 (b)
Find the circular convolution of the following sequences:
\[X_{1}(n) = \{ \underset {\uparrow }{1}, 2,3,4\}\ \ \ \ \ X_{2}(n) \{ \underset {\uparrow }{2}, 1,2,1\}\]
\[X_{1}(n) = \{ \underset {\uparrow }{1}, 2,3,4\}\ \ \ \ \ X_{2}(n) \{ \underset {\uparrow }{2}, 1,2,1\}\]
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4 (c)
Design a Digital low pass FIR filter using Kaiser window to meet the following specifications:
0.99 ≤?H(ejw )≤ 1.01 , 0 ≤w ≤ 0.4π
?H(ejw )?≤ 0.001 , 0.6π ≤ w ?π
0.99 ≤?H(ejw )≤ 1.01 , 0 ≤w ≤ 0.4π
?H(ejw )?≤ 0.001 , 0.6π ≤ w ?π
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4 (d)
Consider the real finite-length sequence x(n).
\[X(n) = \{ \underset {\uparrow }{4}, 3,2,1\}\]
(i)Sketch the finite length sequence y(n) whose six-point DFT is Y(k) = W64k X(k) , Where X(k) is the six-point DFT of x(n).
(ii) Sketch the finite length sequence w(n) whose six-point DFT is W(k) = Re{ X(k) }
(iii) Sketch the finite length sequence q(n) whose three-point DFT is Q(k) = X(2k) , k=0,1,2
\[X(n) = \{ \underset {\uparrow }{4}, 3,2,1\}\]
(i)Sketch the finite length sequence y(n) whose six-point DFT is Y(k) = W64k X(k) , Where X(k) is the six-point DFT of x(n).
(ii) Sketch the finite length sequence w(n) whose six-point DFT is W(k) = Re{ X(k) }
(iii) Sketch the finite length sequence q(n) whose three-point DFT is Q(k) = X(2k) , k=0,1,2
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5 (a)
Explain the Decimation in Time FFT algorithm.
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5 (b)
Discuss the applications of digital signal processing with suitable examples.
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5 (c)
Discuss the key features of the architecture of DSP Processors.
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5 (d)
Write a short note on coefficient quantization in IIR filters.
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