ECET345 Signals and Systems

                                                                                Homework #7

Name of Student _____________________________________

1.       A sine wave of 60 Hz, amplitude of 117 V, and initial phase of zero (or 117 sin(2π*60t) is full wave rectified and sampled at 2,048 samples per second after full wave rectification. Research the Fourier series for a full wave rectified sine wave (on the Internet or in circuit theory books, such as Linear Circuits by Ronald E. Scott) and write it below.

Then write a MATLAB program that samples and stores 4,096 points of full wave rectified sine wave and performs Fourier analysis (FFT) of the full wave rectified sine wave on the stored points.

Plot the results in both linear and log scale (in two separate figures) and extract the amplitude of the DC component and the first four harmonics (first , second, third, and fourth multiple of the fundamental frequency) of the Fourier analysis, then enter them in the table given below. The DC component is given by the first number in the Fourier analysis. Hint: Full wave rectification can be achieved in MATLAB simply by taking the absolute value (abs command) of the sine wave.

Compare the results provided by the Fourier transform and fill out the following table.

2.       Given a cosine wave of frequency (1/π) Hz and amplitude of 10, sampled at 10 samples/second, express it in a complex exponential form.

Hint:

3.       Using Euler’s formula,  ,

(a)    Express  as a complex number, a + jb, and find the numerical values of a and b.

Hint: The arguments of trigonometric functions are in radians and not in degrees.

(b)   Evaluate

and express the result as a complex number in rectangular form. Show the various steps you took in finding the answer. Hint: .

(c)    Using Euler’s formula, show that  is equal to -1. Having proved this, we can consider the nth root of  as the nth root of -1. Thus, using Euler’s formula, find the fourth root of -1 . In other words, find .

4.       A periodic sequence (i.e., a sequence continuing forever) is given by

f(nT) = [ 0011 0000 1100 0011 0000 1100 …….].

(a)    What is its digital period N (i.e., the lowest number of bits beyond which the signal repeats itself)?

(b)   Using the fundamental definition of discrete Fourier transform (DFT), find the numerical values of the first and last DFT terms (of a periodic sequence with digital period of 4, if the first four terms of the sequence are given by 2, 2, -1, and 1.

x(n) = [2  2  -1  1] , N=4        DFT is  4 point

5.       A signal is sampled at 1,024 samples/second and you have collected and stored 32,768 samples after sampling. You apply the FFT algorithm to it, using all of the stored data. What will be the frequency resolution of the FFT in hertz? Hint: See the Week 7 Lecture for the definition of resolution of FFT.

6.       You are required to sample a signal consisting of a mixture of 10, 10.25, and 12 Hz sinusoids that you want to be able to resolve in your Fourier (spectrum) analysis. Your data acquisition hardware can store up to 32,768 samples. Assuming that you store all 32,768 samples, what is the range of sampling frequencies such that you neither violate the Nyquist theorem nor go above the needed resolution?