Download Homework #4 Problems - Communication Systems I | ECE 5660 and more Assignments Digital Communication Systems in PDF only on Docsity! ECE 5660 Spring 2006 Utah State University Homework 4 Due: Wednesday, February 8, 2005 Problems 1. A real-valued signal x(t) is known to be uniquely determined by its samples when the sampling fre- quency is ωs = 2π · 5000 rads/sec. For what values of ω is X(jω) guaranteed to be zero? 2. A continuous-time signal x(t) is obtained at the output of an ideal low pass filter with cutoff frequency ωo = 2π ·500 rads/sec. If impulse-train sampling is performed on x(t), which of the following sampling periods would guarantee that x(t) can be recovered from its sampled version using an appropriate low pass filter? (a) T = 0.5 × 10−3 (b) T = 2 × 10−3 (c) T = 10−4 3. Suppose the sinusoidal signal x(t) = cos(2πft) is sampled at the rate fs Hz. Given the continuous-time frequency f and the sample rate fs, compute the discrete-time frequency after aliasing. (a) f = 2100 Hz, fs = 1600 Hz. (b) f = −2100 Hz, fs = 1600 Hz. (c) f = 1500 Hz, fs = 1600 Hz. (d) f = −4500 Hz, fs = 1600 Hz. (e) Show that cos([π + ]n) = cos([π − ]n) for all n where is a small positive number. 4. A real-valued discrete-time signal x[n] has a Fourier transform X(ejΩ) that is zero for 3π 14 ≤ |Ω| ≤ π. This signal can be up sampled by a factor of L and and then down sampled by a factor of M to give a signal xr[n] for which the spectrum Xr(e jΩ) occupies the entire region |Ω| ≤ π. Specify the values of L and M and sketch the spectra of the signal at each point in the up sampling and down sampling process. 5. The figure below shows the overall system for filtering a continuous-time signal using a discrete-time filter where p(t) = ∑ ∞ n=−∞ δ(t − nT ) and T is the sampling period. If Xc(jω), H(e jΩ) and G(jω) are as shown in the figure with ωM = 2π ·5000 rads/sec and 1 T = 20 kHz, sketch Xp(jω), Xd(e jΩ), Yd(e jΩ), Yp(jω), and Yc(jω). The MAP function converts and impulse train to a discrete-time sequence and MAP−1 does the opposite. MAP−1MAP Xc(jω) ωM−ωM 0 ω 0−π 4 π 4 Ω Hd(e jΩ) 1 G(jω) 0 ω π T − π T T p(t) xc(t) xp(t) xd[n] yd[n] yp(t) yc(t)hd[n] Hd(e jΩ) G(jω) 1 1 ECE 5660 Spring 2006 Utah State University Homework 4 6. (Downsampling by a factor of 2) Consider a discrete-time sequence x[n] from which we form two new sequences, xp[n] and xd[n], where xp[n] corresponds to multiplying x[n] by p[n] = ∑ ∞ k=−∞ δ[n − kN ] and N = 2, and xd[n] is equal to xp[nN ] and N = 2. That is, xd[n] is equal every N th sample of xp[n]. xp[n] = { x[n] n = 0,±2,±4, · · · 0 n = ±1,±3, · · · xd[n] = xp[nN ] = x[nN ] N = 2 (a) If x[n] is the sequence illustrated below, then sketch xp[n] and xd[n]. (b) If X(ejΩ) is as shown below, sketch Xp(e jΩ) and Xd(e jΩ). n Ω X(ejΩ) 2π−2π 3π/4−3π/4 0 x[n] 2
