1. Home
  2. Software receiver design
  3. Software receiver design
  4. The six elements

0.2 The six elements  (Page 8/10)

<< Chapter < Page
  Software receiver design     Page 8 / 10
Chapter >> Page >
The discrete signal x[k] is downsampled by a factor of m by removing all but one of every m samples. The resulting signal is y[i], which takes on values y[i]=x[k] whenever k=im+n.
The discrete signal x [ k ] is downsampled by a factor of m by removing all but one of every m samples. The resulting signal is y [ i ] , which takes on values y [ i ] = x [ k ] whenever k = i m + n .

Mimicking the code in speccos.m with the sampling interval Ts=1/100 , find the spectrum of a cosine wave cos ( 2 π f t ) 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?

The fourth element: static nonlinearities

Linear systems To be accurate, these systems must be exponentially stable and time invariant. such as filters cannot add new frequencies to a signal, though they can remove unwanted frequencies.Nonlinearities such as squaring and quantizing can and will add new frequencies. These can be usefulin the communication system in a variety of ways.

Perhaps the simplest (memoryless) nonlinearity is the square, which takes its input at each time instant and multiplies itby itself. Suppose the input is a sinusoid at frequency f , that is, x ( t ) = cos ( 2 π f t ) . Then the output is the sinusoid squared, which can be rewritten using the cosine-cosine product [link] as

y ( t ) = x 2 ( t ) = cos 2 ( 2 π f t ) = 1 2 + 1 2 cos ( 2 π ( 2 f ) t ) .

The spectrum of y ( t ) has a spike at 0 Hz due to the constant, and a spike at ± 2 f Hz from the double frequency term. Unfortunately, the action of a squaring elementis not always as simple as this example might suggest. The following exercises encourage you to explore thekinds of changes that occur in the spectra when using a variety of simple nonlinear elements.

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

  1. x ( t ) = cos ( 2 π f t ) for f = 100 Hz,
  2. x ( t ) = cos ( 2 π f 1 t ) + cos ( 2 π f 2 t ) 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 ( t ) cannot be greater than that of x ( t ) . 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 ( t ) = | x ( t ) | . Find the spectrum of the output when the input is

  1. x ( t ) = cos ( 2 π f t ) for f = 100 Hz,
  2. x ( t ) = cos ( 2 π f 1 t ) + cos ( 2 π f 2 t ) 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 ( t ) is the sum of three sinusoids f 1 , f 2 , and f 3 ?

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

g ( t ) = 1 x ( t ) > 0 - 1 x ( t ) ≤ 0

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 ( t ) = cos ( 2 π f t ) for f = 100 Hz,
  2. x ( t ) = cos ( 2 π f 1 t ) + cos ( 2 π f 2 t ) for f 1 = 100 and f 2 = 150 Hz.
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Software receiver design' conversation and receive update notifications?

Ask