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Signal Representation by Fourier Series - Problems with Hints | EE 701, Assignments of Electrical and Electronics Engineering

Material Type: Assignment; Class: Linear Systems; Subject: Electrical Engineering; University: Wright State University-Main Campus; Term: Fall 2008;

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EGR/EE 701 Linear Systems Fall 2008
Chapter 3: Signal representation By Fourier Series Week 03
Problems: 3.1-2, 3.2-1, 3.3-1, 3.4-1M, 3.4-3M, 3.4-10M, 3.5-1M, 3.5-2, 3.5-6
3.1-2 (a) For the signals f(t) and x(t) depicted in Fig. P3.1-2, find the component of the form x(t) contained in f(t). In other
words find the optimum value of c in the approximation f(t) cx(t) so that the error signal energy is minimum.
(b) Find the error signal e(t) and its energy E e. Show that the error signal is orthogonal to x(t), and that E f = c 2 E x + E e.
Can you explain this result in terms of vectors.
Fig. P3.1-2
3.2-1 Find the correlation coefficient c n of signal x(t) and each of the four pulses f1(t), f2(t), f3(t), and f4(t) depicted in Fig.
P3.2-1. Which pair of pulses would you select for a binary communication in order to provide maximum margin against
noise along the transmission path?
Fig. P3.2-1
3.3-1 Let x1(t) and x2(t) be two signals orthonormal (that is, with unit energies) over an interval from t = t1 to t2. Consider a
signal f(t) where
f(t) = c1 x1 (t) + c2 x2(t) t1 t t2
This signal can be represented by a two-dimensional vector f(c1, c2).
(a) Determine the vector representation of the following six signals in the two-dimensional vector spsce:
(i) f1(t) = 2 x1(t)
x2(t) (ii) f2(t) = x1(t) + 2 x2(t)
(iii) f3(t) = x2(t) (iv) f4(t) = x1(t) + 2 x2(t)
(v) f5(t) = 2 x1(t) + x2(t) (vi) f6(t) = 3 x1(t)
(b) Point out pairs of mutually orthogonal vectors among these six vectors. Verify that the pairs of signals corresponding to
these orthogonal vectors are also orthogonal.
3.4-1 (a) Sketch the signal f(t) = t 2 for all t and find the trigonometric Fourier series ϕ(t) to represent f(t) over the interval ( 1,
1). Sketch ϕ(t) for all values of t.
3.4-3 For each of the periodic signals shown in Fig. P3.4-3, find the compact trigonometric Fourier series and sketch the
amplitude and phase spectra. If either the sine or cosine terms are absent in the Fourier series, explain why.
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EGR/EE 701 Linear Systems Fall 2008

Chapter 3 : Signal representation By Fourier Series Week 03

Problems: 3.1-2, 3.2-1, 3.3-1, 3.4-1M, 3.4-3M, 3.4-10M, 3.5-1M, 3.5-2, 3.5-

3.1-2 (a) For the signals f ( t ) and x ( t ) depicted in Fig. P3.1-2 , find the component of the form x ( t ) contained in f ( t ). In other words find the optimum value of c in the approximation f ( t ) ≈ cx ( t ) so that the error signal energy is minimum. (b) Find the error signal e ( t ) and its energy E (^) e_._ Show that the error signal is orthogonal to x ( t ), and that E (^) f = c

2 E (^) x + E (^) e_._ Can you explain this result in terms of vectors.

Fig. P3.1-

3.2-1 Find the correlation coefficient c (^) n of signal x ( t ) and each of the four pulses f 1 ( t ), f 2 ( t ), f 3 ( t ), and f 4 ( t ) depicted in Fig. P3.2-1. Which pair of pulses would you select for a binary communication in order to provide maximum margin against noise along the transmission path?

Fig. P3.2-

3.3-1 Let x 1 ( t ) and x 2 ( t ) be two signals orthonormal (that is, with unit energies) over an interval from t = t 1 to t 2. Consider a signal f ( t ) where

f ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) t 1 ≤ tt 2

This signal can be represented by a two-dimensional vector f ( c 1 , c 2 ). (a) Determine the vector representation of the following six signals in the two-dimensional vector spsce: (i) f 1 ( t ) = 2 x 1 ( t ) −−−− x 2 ( t ) (ii) f 2 ( t ) = − x 1 ( t ) + 2 x 2 ( t ) (iii) f 3 ( t ) = − x 2 ( t ) (iv) f 4 ( t ) = x 1 ( t ) + 2 x 2 ( t ) (v) f 5 ( t ) = 2 x 1 ( t ) + x 2 ( t ) (vi) f 6 ( t ) = 3 x 1 ( t ) (b) Point out pairs of mutually orthogonal vectors among these six vectors. Verify that the pairs of signals corresponding to these orthogonal vectors are also orthogonal.

3.4-1 (a) Sketch the signal f ( t ) = t

2 for all t and find the trigonometric Fourier series ϕ( t ) to represent f ( t ) over the interval (− 1, 1). Sketch ϕ( t ) for all values of t.

3.4-3 For each of the periodic signals shown in Fig. P3.4-3 , find the compact trigonometric Fourier series and sketch the amplitude and phase spectra. If either the sine or cosine terms are absent in the Fourier series, explain why.

Fig. P3.4-

3.4-10 Find the trigonometric Fourier series for f ( t ) shown in Fig. P3.4-10 over the interval [0, 1]. Use ω 0 = 2 π. Sketch the Fourier series ϕ( t ) for all t. Compute the energy of the error signal e ( t ) if the number of terms in the Fourier series are N for N = 1, 2, 3, and 4. Hint: Use Eq. (3.40) to compute error energy.

Fig. P3.4-10.

3.5-1 For each of the periodic signals in Fig. P3.4-3, find the exponential Fourier series and sketch the corresponding spectra.

3.5-2 The trigonometric Fourier series of a certain periodic signal is given by

f ( t ) = 3 + 3 cos 2 t + sin 2 t + sin 3 t −−−− (1/2) cos (5 t + [π/3])

(a) Sketch the trigonometric Fourier spectra. (b) By inspection of the spectra in part a, sketch the exponential Fourier series spectra. (c) By inspection of the spectra in part b, write the exponential Fourier series for f ( t ). Hint: To express the Fourier series in compact form, combine the sine and cosine terms of the same frequency. Moreover, all terms must appear in the cosine form with positive amplitudes. This can always be done by suitably adjusting the phase.

3.5-6 (a) The Fourier series for the periodic signal in Fig. 3.10b is given in Exercise E3.6. Verify Parseval’s theorem for this series, given that