2. A signal r(t) is modulated with a carrier signal with frequency fe, but demodulated slightly out of tune with a frequency fc + Afc What will X(f) and r(t) look like, after suitable filtering? You may solve this by using trigonometric relations like cos(a - b) = cos(a) cos(b) + sin(a) sin(b) (you can get this all from WolframAlpha). You may also solve this in the frequency domain, by considering: • The Fourier transform of the product of functions is the convolution of the Fourier transforms of the functions, (you know this in the reverse case) • That the Fourier transform of the cos function is a sum of delta functions (you knew that), • The sampling property of delta functions (you know those, too).

Part 4: Is there any visible light source that would make you perceive the can as being bright with dark letters (i.e., that makes the lettering on the can appear to be darker than the red background of the can)? Answer yes or no.

Given the following function, F(s)=\frac{4}{1+0.02 s+0.01 s^{2}}, Show that its poles are complex conjugate. Calculate the undamped natural frequency wo and the damping ratio of the poles.

c)the measured intensity for soil is highest in which visible band? A) redB)greenC)blue

3.11. Suppose we are given the following information about a signal x[n]: 1. x[n] is a real and even signal. 2. x[n] has period N = 10 and Fourier coefficients ak. 3. a11 = 5. \text { 4. } \frac{1}{10} \sum_{n=0}^{9}|x[n]|^{2}=50 Show that x[n] = A cos(Bn + C), and specify numerical values for the constants A,B, and C.

6) Find the zero-state response for the following. \begin{array}{l} h[n]=(0.3)^{n} u[n] \\ x[n]=(0.1)^{n} u[n] \end{array}

Question 4: Cascaded Gaussians (10 pts) Define the row vector G = [1 1], which we claim approximates an unnormalized Gaussian smoothing kernel with a standard deviation of 1 / 2. Also, define the notation |G|^n to mean kernel Gconvolved with itself n-1 times. That is, |G|^2 = G*G, and |G|^3 = G*G*G, and soon. on.A) (8 pts) Fill in the values for the blanks in the following table. If the value is an nteger, enter an integer value, otherwise round to 2 decimal places.

2. Windows and Zero Padding. Download the file HW6Problem2.mat from CANVAS. The file has a signal x and sampling frequency fs. The signal contains two sinusoids with frequencies in the range of 190 -210 Hz and some noise. You will analyze the signal in multiple ways and try to determine the two frequencies as accurately as possible. Make all plots on 1 figure using the subplot command. a) Plot the fft of the signal using a rectangular window and no zero padding. This basically means X=fft (x). Make your plot range from 190 – 210 Hz. b) Plot the fft of the signal using a rectangular window and zero padding to make the fft have a length of 4096. This means X=fft (x, 4096). Make your plot range from 190-210 Hz. Hint you will need a new frequency vector. c) Plot the fft of the signal using a Hanning window and no zero padding. This basically means X=fft (x.*hanning (length(x)). Make your plot range from 190 – 210 Hz. d) Plot the fft of the signal using a Hanning window and zero padding to make the fft have a length of4096. Make your plot range from 190-210 Hz. e) What is your best estimate of the frequencies of the two signals? f) Which plot was above was the most helpful in identifying the two frequencies? Why?

3. (10 pt) The input to a subwoofer loudspeaker is to pass through a passive low-pass Butterworth filter having a cutoff frequency of 100 Hz. Specify (Design) a filter that meets the following specifications: At 50 Hz, the signal magnitude ratio should be at least at 0.95; at 200 Hz, the magnitude ratio should be no more than 0.08. The sensor and load resistances are 10 Q. Determine the minimum filter stage that satisfies the above specifications and the values for the components. Draw the corresponding electric circuit using capacitor, inductor, and resistance.

E. (1 points) If it was properly normalized, which of the two convolutions kernels used above, A or C, would be smoothing the image?