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K Vasudevan Faculty of EE IIT Kanpur (vasu@iitk.ac.in)  1 EE320ATutorial 1 Date: 9th Aug. 20191. Prove the following properties of the complex exponential Fourier seriesrepresentation, for a real-valued periodic signal  g  p ( t ):(a) If the periodic function  g  p ( t ) is even, that is,  g  p ( − t ) =  g  p ( t ), thenthe Fourier coeﬃcient  c n  is purely real and an even function of   n .(b) If   g  p ( t ) is odd, that is,  g  p ( − t ) = − g  p ( t ), then  c n  is purely imaginaryand an odd function of   n .(c) If   g  p ( t ) has half-wave symmetry, that is,  g  p ( t ± T  0 / 2) = − g  p ( t ), where T  0  is the period of   g  p ( t ), then  c n  consists of only odd order terms. 1 Ω A v 1 ( t ) Bv i ( t ) = 3cos(6 πt )(volts)2 F i ( t )  ddt v o ( t ) =  dv 1 ( t ) /dt + − Figure 1:  Block diagram of the system. 2. Consider the system shown in Figure 1. Assume that the current in branch AB  is zero. The voltage at point  A  is  v 1 ( t ).(a) Find out the time domain expression that relates  v 1 ( t ) and  v i ( t ).(b) Using the Fourier transform, ﬁnd out the relation between  V  1 ( f  ) and V  i ( f  ).(c) Find the relation between  V  o ( f  ) and  V  i ( f  ).(d) Compute the power when the output voltage  v o ( t ) is applied acrossa 1 ohm resistor.3. Consider a complex-valued signal  g ( t ). Let  g 1 ( t ) =  g ∗ ( − t ). Let  g ( n )1  ( t )denote the  n th derivative of   g 1 ( t ). Consider another signal  g 2 ( t ) =  g ( n ) ( t ).Is  g ∗ 2 ( − t ) =  g ( n )1  ( t )? Justify your answer using Fourier transforms.4. Prove the following using the Schwarz’s inequality: | G ( f  ) | ≤    ∞ t = −∞ | g ( t ) |  dt |  j2 πfG ( f  ) | ≤    ∞ t = −∞  dg ( t ) dt  dt  (j2 πf  ) 2 G ( f  )  ≤    ∞ t = −∞  d 2 g ( t ) dt 2  dt.  K Vasudevan Faculty of EE IIT Kanpur (vasu@iitk.ac.in)  2 g ( t ) t − 2  − 1 1 21 Figure 2:  Plot of   g ( t ). The Schwarz’s inequality states that    x 2 x = x 1 f  ( x ) dx  ≤    x 2 x = x 1 | f  ( x ) |  dx. Evaluate the three bounds on  | G ( f  ) |  for the pulse shown in Figure 2. 2 δ ( t )0  T/ 2  T δ ( t − 3 T/ 2) t Figure 3:  A periodic waveform  g p ( t ). 5. Consider the periodic waveform  g  p ( t ) given in Figure 3, where  T   denotesthe period.(a) Compute the real Fourier series representation of   g  p ( t ). Give theexpression for the coeﬃcient of the  n th term.(b) Compute the Fourier transform of   g  p ( t ).6. A nonlinear system deﬁned by y ( t ) =  x ( t ) + 0 . 2 x 2 ( t )has an input signal with a bandpass spectrum given by X  ( f  ) = 4rect  f   − 206   + 4rect  f   + 206  . Sketch the spectrum at the output labelling all the important frequenciesand amplitudes.

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Sep 22, 2019

Sep 22, 2019
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