Post on 05-May-2018
UNIT 4: FOURIER REPRESENTATION FOR SIGNALS – 11. State and prove the periodic time shift and periodic properties of DTFS.
[ June 14 6marks]
Solution:
Periodicity:
As a consequence of Eq. (6.41), in the discrete-time case we have to consider values of
R(radians) only over the range0 < Ω < 2π or π < Ω < π, while in the continuous-time case we have to consider values of 0 (radians/second) over the entire range –∞ < ω < ∞.
Time Shifting:
2. Find the inverse Fourier transform x [ n ] [ Jan14, 6 marks]
Solution:
3. State and prove the time shift and periodic time convolution properties of DTFS. [June 13, 6marks]
Solution:
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Signals and Systems 10EE52
4. Evaluate the DTFS representation for the signal, x[n]= Sin(4πn/21) + cos(=10 πn/21) + 1.
Sketch. the magnitude and phase spectra. [June 13, 8marks]
Solution:
5.Prove the following properties[Jan15, 8marks]
i Convolution property of periodic discrete time sequences
ii Parsevals relation for the FS
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Signals and Systems 10EE52
Solution:
6.Determine the FS representation for the signal x(t) of fundamental period T given
by . Sketch the magnitude and phase of X(k). [Jan15, 8marks]
Solution:
7. State and prove the convolution property of fourier series. [June 15, 6marks]
Solution:
Convolution:
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Signals and Systems 10EE52
As in the case of the z-transform, this convolution property plays an important role in the study of discrete-time LTI systems.
Duality:
The duality property of a continuous-time Fourier transform is expressed as
There is no discrete-time counterpart of this property. However, there is a duality between the discrete-time Fourier transform and the continuous-time Fourier series. Let
8. Evaluate the DTFS representation for the signal.
Sketch the magnitude and phase spectra.[June 13, 8marks]
Solution:
9. Determine the Fourier series for the signal [June 13, 6marks]
Solution: