Download Continuous Time Signal Analysis - The Fourier Transform | EE 701 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! 1 EGR/EE 701 Linear Systems Fall 2008 Chapter 4: Continuous Time Signal Analysis: The Fourier Transform Week 04 Problems: 4.1-7M, 4.2-4M, 4.3-2M, 4.3-6, 4.3-9, 4-3.10, 4.4-3, 4.5-3, 4.7-1 4.1-7 Using Eq. (4.8b), find the inverse Fourier transforms of the spectra in Fig. P4.1-7. Fig. P4.1-7 4.2-4 Find the inverse Fourier transform of F(ω) for the spectra illustrated in Figs. P4.2-4a and b. Hint: F(ω) = | F(ω) | e j ∠ F(ω). This problem illustrates how different phase spectra (both with the same amplitude spectrum) represent entirely different signals. Fig. P4.2-4 4.3-2 The Fourier transform of the triangular pulse f(t) in Fig. P4.3-2a is expressed as F(ω) = (1/ω2) [ e j ω − j e j ω − 1] Using this information, and the time-shifting and time-scaling properties, find the Fourier transforms of the signals f i (t) (i = 1, 2, 3, 4, 5) shown in Fig. P4.3-2. Hint: See Sec. 1.3 for explanation of various signal operations. Pulses f i (t) (i = 2, 3, 4 can be expressed as a combination of f(t) and f1(t) with suitable time shift (which may be positive or negative). 2 Fig. P4.3-2 4.3-6 The signals in Fig. P4.3-6 are modulated signals with carrier cos 10t. Find the Fourier transforms of these signals using the appropriate properties of the Fourier transform and Table 4.1. Sketch the amplitude and phase spectra for parts (a) and (b). Fig. P4.3-6 4.3-9 A signal f(t) is bandlimited to B Hz. Show that the signal f n (t) is bandlimited to nB Hz. Hint: Start with n = 2. Use frequency convolution property and the width property of convolution. 4.3-10 Find the Fourier transform of the signal in Fig. P4.3-3a by three different methods: (a) By direct integration using the definition (4.8a). (b) Using only pair 17 Table 4.1 and the time-shifting property. (c) Using the time-differentiation and time-shifting properties, along with the fact δ(t) ⇔ 1 Hint: 1 − cos 2x = 2 sin 2 x 4.4-3 Signals f1(t) = 10 4 rect (10 4 t) and f2(t) = δ(t) are applied at the inputs of the ideal lowpass filters H1(ω) = rect (ω/40000 π) and H2(ω) = rect(ω/20000π) (Fig. P4.4-3). The outputs y1(t) and y2(t) of these filters are multiplied to obtain the signal y(t) = y1(t) y2(t). (a) Sketch F1(ω) and F2(ω) . (b) Sketch H1(ω) and H2(ω) (c) Sketch Y1(ω) and Y2(ω). (d) Find the bandwidths of y1(t) , y2(t), and y(t) . Hint for part (d): Use the convolution property and the width property of convolution to determine the bandwidth of y1(t) y2(t).
