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.
