1) For signal x(t) below, generate the corresponding graph using the following operation: y(t) = x(2t - 4)

14.2 A sinusoid with 1-V peak amplitude is applied at the input of a filter having the transfer function Find the peak amplitude and the phase (relative to that of the input sinusoid) of the output sinusoid if the frequency of the input sinusoid is (a) 1 kHz, (b) 10 kHz, (c) 100 kHz, and (d) 1 MHz.

Problem 1 [15 points] Determine the Fourier Transform of the following waveform for A = 12 and T = 3 s.

Problem 2 [15 points] For the given signals 1. r(t) = 4 sin(20πt) 2. r₂(t) 4 sin(80nt) 3. T3(t) = -4 sin(40πt) Answer the following questions a. [6 points] Using the MATLAB command stem to plot the signals formed by sampling the above three functions over the interval 0 < t <1 with sampling rate of 30 Hz. b. [6 points] Determine the Nyquist sampling rate for each signal. c. [3 points] What do you observe?

Problem 3 [15 points] A commonly used method for converting discrete-time signals to continuous- time signals is zero-order hold, in which z(t) is held constant between its known samples at t = nTg. Zero-order hold interpolation of samples is illustrated in the following figure a. [10 points] Zero-order hold is a low pass filter. Compute its frequency response H(w). Hint: The sampled signal is z, (t) = a[n]d(t-nT) and the zero-order hold signal is equivalent to convolving the sample signal r(t) with a rectangular pulse of with T. b. [5 points] Zero-order hold is used to reconstruct cos(0.1t/T.) from its samples at t = nT,. The first copy of the spectrum induced by sampling lies in

Problem 4 [20 points] Use MATLAB command tf and bode, generate magnitude (in decibel) and phase plots (in degree) for the following voltage transfer functions.

The continuous time signal xc(t)=cos(49.7mt). is sampled with a sample period T. z[n]=cos(in),-oo<n<oo where A=9 Choose the smallest possible value of T in milliseconds/sample consistent with this information. Provide a number as your answer with an accuracy of two decimal digits.

Problem 6 [25 points] What does aliasing sound like? Load the file P673.mat from the Canvas. This is a speech signal (a single sentence) sampled at 24000 samples/s. a. [5 points] Listen to the signal using Describe what you hear. b. [5 points] Plot the one-sided magnitude spectrum from 0 to 8 kHz using N-length (X)/3; F=linspace (0,8000,N); FX=abs(fft (X)); plot (F,FX(1:N)) c. [5 points] Repeat (a) and (b) after reducing the sampling rate to 6000 samples/s. Do this by keeping only every fourth sample and discarding the other three samples. Use Y=X(1:4:end); soundsc (Y,6000) Describe what you hear. It should sound different. d. [5 points] Plot the one-sided magnitude spectrum of the signal in (c) from 0 to 3 kHz using Nl-length (Y)/2; Fl=linspace (0,3000, N1); FY=4*abs (fft (Y)); plot (F1,FY(1:N1)) e. [5 points] Compare (note differences) answers to (a) and (c), and to (b) and (d).

1. State whether the given systems are stable, marginally stable or unstable. Also explain why. A) (D²+3D+2)D²y(t)=(D+3)x(t) B) (D+3)y(t)=(D²+D+1)x(t) C) (D²+D+1) (D²+D+1)y(t)=Dx(t) D) (D²+5D+4)y(t)= (D²+D+1)x(t) E) (D²-5D+4)y(t)= (D+5)x(t)

The following simultaneous ordinary differential equations present a mathematical model for a system: