STUPIDSID
1 (a)
Compute the DFT of the sequence \[x(n)=\cos\left ( \dfrac{n\pi}{4} \right )\] for N=4, plot |x(k)| and ∠x(k).
9 M
1 (b)
Find the DFT of the sequence x(n)=0.5n u(n) for 0 < n <3 by evaluating x(n)=an for 0< n < N - 1.
7 M
1 (c)
Find the relation between DFT and Z transform.
4 M
2 (a)
State and prove the linearity property of DFT and symmetrical property.
5 M
2 (b)
The five samples of the 8 point DFT x(k) are given as
x(0)=0.25, x(1)=1.25 - j0.3018, x(6)=x(4)=0, x(5)=0.125 - j0.0518.
x(0)=0.25, x(1)=1.25 - j0.3018, x(6)=x(4)=0, x(5)=0.125 - j0.0518.
5 M
2 (c)
For x(n)={1,-2,3,-4,5-6}, without computing its DFT, find the following.
\[i)\ x(o)\ ii)\ \sum_{k=0}^{5}\ iii)\ X(3)\ iv)\ \sum_{k=0}^{5}1 \times(k)|^{2}\ v)\ \sum_{k=0}^{5}(-1)^{k}\times (k)\]
\[i)\ x(o)\ ii)\ \sum_{k=0}^{5}\ iii)\ X(3)\ iv)\ \sum_{k=0}^{5}1 \times(k)|^{2}\ v)\ \sum_{k=0}^{5}(-1)^{k}\times (k)\]
10 M
3 (a)
Consider a FIR filter with impulse response
h(n)={1,1,1}, if the input is
X(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}. Find the output y(a) using overlap add method.
h(n)={1,1,1}, if the input is
X(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}. Find the output y(a) using overlap add method.
12 M
3 (b)
What is an plane computation? What is total number of complex additions and multiplication required for N=256 point, if DFT is computed directly and if FFT is used?
3 M
3 (c)
For sequence x(n)={2,0,2,0} determine x(2) using Goertzel Filter. Assume the zero initial conditions.
5 M
4 (a)
Find the circular convolution of x(n)={1,1,1,1} and h(n)={1,0,1,0} using DIF-FFT algorithm.
12 M
4 (b)
Derive DIT-FFT algorithm for N=4. Draw the complete signal How graph?
8 M
5 (a)
Design a Chebyshev analog filter (low pass) that has a-3dB cut-off frequency of 100 rad/sce and a stopband attenuation 25dB or greater for all radian frequencies past 250 rad/sec.
14 M
5 (b)
Compare Butterworth and Chebyshev filters.
3 M
5 (c)
Let \[H(s)=\dfrac{1}{s^{2}+s+1}\] represent the transfer function of LPF with a passband of 1 rad/sec. Use frequency transformation (Analog to Analog) to find the transfer function of a band pass filter with passband 10 rad/sec and a centre frequency of 100 rad/sec.
3 M
6 (a)
Obtain block diagram of the direct form I and direct from II realization for a digital IIR filter described by the system function.
\[H(z)=\dfrac{8z^{3}-4z^{2}+11z-2}{\left ( z-\dfrac{1}{4} \right )\left ( z^{2}-z+\dfrac{1}{2} \right )}\]
\[H(z)=\dfrac{8z^{3}-4z^{2}+11z-2}{\left ( z-\dfrac{1}{4} \right )\left ( z^{2}-z+\dfrac{1}{2} \right )}\]
10 M
6 (b)
Find the transfer function and difference equation realization shown in fig Q6(b)
6 M
6 (c)
Obtain the direct form realization of liner phase FIR system given by
\[H(z)=1+\dfrac{2}{3}z^{-1}+\dfrac{15}{8}z^{-2}\]
\[H(z)=1+\dfrac{2}{3}z^{-1}+\dfrac{15}{8}z^{-2}\]
4 M
7 (a)
"The desired frequency response of low pass filter is given by
\[H_{d}(e^{jw})=H_{d}(\infty )=\left\{\begin{matrix} e^{-j3w} &|\infty|\dfrac{3 \pi}{4} \\0 &\dfrac{3\pi}{4}<|\infty|<\pi \end{matrix}\right\]
Determine the frequency response of the FIR if Hamming window is used with N=7."
\[H_{d}(e^{jw})=H_{d}(\infty )=\left\{\begin{matrix} e^{-j3w} &|\infty|\dfrac{3 \pi}{4} \\0 &\dfrac{3\pi}{4}<|\infty|<\pi \end{matrix}\right\]
Determine the frequency response of the FIR if Hamming window is used with N=7."
10 M
7 (b)
Compare IIR filter and FIR filters.
6 M
7 (c)
Consider the pole-zero plot as shown in FigQ7(c). i) Does it represent an FIR filter? ii) Is it linear phase system?
4 M
8 (a)
Design a digital filter H(z) that when used in an A/D-H(z)-D/A structure gives an equivalent analog filter with the following specification:
Passband ripple:≤3.01dB
Passband edge : 500Hz
Stopband attenuation : ≥ 15dB
Stopband edge : 750 Hz
Sample Rate : 2 KHz
Use Bilinear transformation to design the filter on an analog system function. Use Butterworth filter prototype. Also obtain the difference equation.
Passband ripple:≤3.01dB
Passband edge : 500Hz
Stopband attenuation : ≥ 15dB
Stopband edge : 750 Hz
Sample Rate : 2 KHz
Use Bilinear transformation to design the filter on an analog system function. Use Butterworth filter prototype. Also obtain the difference equation.
14 M
8 (b)
Transform the analog filter
\[H_{a}(s)=\dfrac{s+1}{s^{2}+5s+6}\]
Into H(z) using impulse invariant transformation Take T=0.1 Sec.
\[H_{a}(s)=\dfrac{s+1}{s^{2}+5s+6}\]
Into H(z) using impulse invariant transformation Take T=0.1 Sec.
6 M
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