I 1 3 [ (i) (ii) (iii) (iv) (v) (vi) Generate L=100 samples of a discrete-time cosine wave with A =5, p = /5 and w=0.25 with the following equation: x[n] = Acos(an+q) for n = 0, 1, 2,..., L- 1 Store this signal in the vector xx, which can also be used in succeeding parts. Now, use firfilt() to implement the following filter on the signal xx. y[n] 10x[n] 10x[n-1] (6) This is called a first-different filter, but with a gain of ten. In Matlab, you must define the vector bb needed in firfilt. (vii) (5) (viii) Note that y[n] and x[n] are different lengths. What is the length of the filtered signal, and why is it that length? Using subplot, plot the first 100 samples of both waveforms x[n] and y[n] on the exact figure. Use the stem function to make a discrete-time signal plot, but label the x-axis to run over the range 0 n 100. Verify the amplitude and phase of x[n] directly from its plot in the time domain. From the plot, except for the first sample y[0], the sequence y[n] seems to be a scaled and shifted cosine wave of the same frequency as the input. Explain why the first sample is different from the others. Determine the frequency, amplitude, and phase of y[n] directly from the plot. Ignore the first output point, y[0]. Characterize the filter performance at the input frequency by computing the relative amplitude and phase, i.e., the ratio of output to input amplitudes and the difference between output and input phases. Derive the mathematical expression for the output when the input signal is a complex exponential x[n] = exp(jnw). From this formula, determine how much the amplitude and phase should change for x[n], which has a frequency of w=0.5 , and compare with your measured results above.