Solution Manual for Fundamentals of Digital Signal Processing Using MATLAB 2nd Edition by Schilling

1.1 Suppose the input to an amplifier is x a (t) = sin(2πF 0 t) and the steady-state output is y a (t) = 100 sin(2πF 0 t + φ 1) − 2 sin(4πF 0 t + φ 2) + cos(6πF 0 t + φ 3) (a) Is the amplifier a linear system or is it a nonlinear system? (b) What is the gain of the amplifier? (c) Find the average power of the output signal. (d) What is the total harmonic distortion of the amplifier? Solution (a) The amplifier is nonlinear because the steady-state output contains harmonics. (b) From (1.1.2), the amplifier gain is K = 100. (c) From (1.2.4), the output power is P y = d 2 0 4 + 1 2 d 2 1 + d + 2 2 + d 2 3 = .5(100 2 + 2 2 + 1) = 5002.5 (d) From (1.2.5) THD = 100(P y − d 2 1 /2) P y = 100(5002.5 − 5000) 5002.5 = .05%Solution Manual for Fundamentals of Digital Signal Processing Using MATLAB 2nd Edition by Schilling. Full file at https://testbanku.eu/

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In this paper a mathematical analysis to extract a nonlinear amplifier model from measured characteristics was developed and verified on a case study. Two methods were proposed. The first can be used to extract a fifth order polynomial model by inserting a sine wave at the amplifier input and measuring the output harmonics at different input power levels. It can be used to estimate other nonlinear behaviors of the amplifier; such as intermodulation and desensitization. The other model can be extracted by measuring the two-tone intermodulation spectrum and deduce other amplifier characteristics. Both methods were mathematically analyzed and experimentally verified.

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For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 sin(8πt + 3π/4) = 10cos(4πt + π/8) + 6 cos(8πt + 3π/4 − π/2) = 10cos(4πt + π/8) + 6 cos(8πt + π/4) For the double-sided spectra, write the signal in terms of complex exponentials using Euler's theorem: x(t) = 5exp[(4πt + π/8)] + 5 exp[−j(4πt + π/8)] +3 exp[j(8πt + 3π/4)] + 3 exp[−j(8πt + 3π/4)] The two sets of spectra are plotted in Figures 2.1 and 2.2. Problem 2.2 The result is x(t) = 4e j(8πt+π/2) + 4e −j(8πt+π/2) + 2e j(4πt−π/4) + 2e −j(4πt−π/4) = 8cos(8πt + π/2) + 4 cos (4πt − π/4) = −8 sin (8πt) + 4 cos (4πt − π/4)

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IEEE Microwave and Wireless Components Letters

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Exercise 1.1: I/O equation Consider a second-order filter with the I/O equation: (a) From the I/O equation, calculate the transfer function H(z). (b) From the coefficients of this transfer function (vectors: b = [b0 b1 b2] and a = [1 a1 a2]), write a MATLAB program to compute the first 100 samples of the impulse response, using the filter function. (c) From the impulse response, complete your program to compute the magnitude (in dB) and phase (in °) spectra of the filter, using the fft, abs and angle functions. (d) Display your results in a four quadrant figure (see template below) using the subplot function. To plot the pole/zero pattern, use the following code: (e) Modify your program to set the coefficient a1 value to -1.9 and display the corresponding four quadrant figure. Give your comments on the pole/zero pattern and impulse response. Exercise 2.1: Highpass filter Pole/zero placements are useful to design simple filters. Consider a first-order highpass filter whose the general transfer function is where the coefficients a and b are positive and less than one. The gain G is fixed so as to normalize the filter to unity at the highest frequency. Let’s define the parameter A as the attenuation of the lowest frequency relative to the highest one. Additionally, a constraint is imposed on the filter speed response1 defined by two parameters: eff n (number of samples to reach the steady state) and  (steady state threshold in %). (a) From H(z) , determine the frequency response H(w) where .. is the digital frequency. (b) From H(w) , determine the attenuation A as function of a and b. (c) Determine the coefficient b as function of A and a. (d) Determine the coefficient a as function of .. and eff n. (e) Determine the normalization gain factor G as function of a and b. (f) Write a MATLAB function HighPassFilter_NU to implement the filter, taking . dB eff as input parameters and b, a, G as output parameters. (g) Write a MATLAB program to test your function with the input parameter values . (h) Modify your program to plot the magnitude spectra based on a collection of eff n values. Test it with . (i) Explain how the value eff n modifies the magnitude spectrum . Exercise 2.2: Resonator filter Consider the transform function of a normalized resonator filter . (a) Express the filter coefficients (G, 1a and 2a) in terms of sampling frequency sf, peak frequency 0 f and width . (b) Write a MATLAB function ResonatorFilter_NU that returns the filter coefficients using sf, . and f, as input parameters. (c) Test your function with 30MHz, 2MHz, 0.5MHz s and plot the filter frequency response (magnitude spectrum in dB) over 0 < f

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