0.2 The six elements  (Page 8/10)

Mimicking the code in speccos.m with the sampling interval Ts=1/100 , find the spectrum of a cosine wave $cos\left(2\pi ft\right)$ when f=30, 40, 49, 50, 51, 60 Hz. Which of these show aliasing?

Create a cosine wave with frequency 50 Hz. Plot the spectrum when this wave is sampled at Ts=1/50, 1/90, 1/100, 1/110 , and 1/200 . Which of these show aliasing?

Mimic the code in speccos.m with sampling interval Ts=1/100 to find the spectrum of a square wave with fundamental f=10, 20, 30, 33, 43 Hz. Can you predict where the spikes will occur in each case?Which of the square waves show aliasing?

Mimic the code in speccos.m with Ts=1/1000 to find the spectrum of the output $y\left(t\right)$ of a squaring block when the input is

1. $x\left(t\right)=cos\left(2\pi ft\right)$ for $f=100$ Hz,
2. $x\left(t\right)=cos\left(2\pi f_{1}t\right)+cos\left(2\pi f_{2}t\right)$ for $f_1=100$ and $f_2=150$ Hz,
3. a filtered noise sequence with nonzero spectrum between $f_1=100$ and $f_2=300$ Hz. Hint: generate the input by modifying filternoise.m .
4. Can you explain the large DC (zero frequency) component?

TRUE or FALSE: The bandwidth of $x^4\left(t\right)$ cannot be greater than that of $x\left(t\right)$ . Explain.

Try different values of $f_1$ and $f_2$ in [link] . Can you predict what frequencies will occur in the output.When is aliasing an issue?

Repeat Exercise  [link] when the input is a sum of three sinusoids.

Suppose that the output of a nonlinear block is the rectification (absolute value) of the input $y\left(t\right)=\left|x\right(t\left)\right|$ . Find the spectrum of the output when the input is

1. $x\left(t\right)=cos\left(2\pi ft\right)$ for $f=100$ Hz,
2. $x\left(t\right)=cos\left(2\pi f_{1}t\right)+cos\left(2\pi f_{2}t\right)$ for $f_1=100$ and $f_2=125$ Hz.
3. Repeat (b) for $f_1=110$ and $f_2=200$ Hz. Can you predict what frequencies will be present for any $f_1$ and $f_2$ ?
4. What frequencies will be present if $x\left(t\right)$ is the sum of three sinusoids $f_1$ , $f_2$ , and $f_3$ ?

Suppose that the output of a nonlinear block is $y\left(t\right)=g\left(x\right(t\left)\right)$ , where

is a quantizer that outputs positive one when the input is positive and outputs minus one when the input is negative.Find the spectrum of the output when the input is

1. $x\left(t\right)=cos\left(2\pi ft\right)$ for $f=100$ Hz,
2. $x\left(t\right)=cos\left(2\pi f_{1}t\right)+cos\left(2\pi f_{2}t\right)$ for $f_1=100$ and $f_2=150$ Hz.