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: