Communications Toolbox

Representing Analog Signals

To perform baseband modulation of an analog signal using this toolbox, start with a real message signal and a sampling rate Fs in hertz. For modulation techniques other than quadrature amplitude modulation (QAM), represent the signal using a vector x, the entries of which give the signal's values in time increments of 1/Fs. Baseband modulation (using a technique other than QAM) produces a complex vector.

For example, if t measures time in seconds, then the vector x below is the result of sampling a frequency-one sine wave 100 times per second for 2 seconds. The vector y represents the modulated signal. The output shows that y is complex.

• Fs = 100; % Sampling rate is 100 samples per second.
  t = [0:1/Fs:2]'; % Sampling times for 2 seconds
  x = sin(2*pi*t); % Representation of the signal
  y = amodce(x,Fs,'pm'); % Modulate x to produce y.
  whos
    Name      Size           Bytes  Class
  
    Fs        1x1                8  double array
    t       201x1             1608  double array
    x       201x1             1608  double array
    y       201x1             3216  double array (complex)
  
  Grand total is 604 elements using 6440 bytes
  

Baseband Modulated Signals Defined

This section explains the connection between this complex vector y and the real signal that you might expect to get after modulating a real signal. If the modulated signal has the waveform

where f_c is the carrier frequency and is the carrier signal's initial phase, then a baseband simulation recognizes that this equals the real part of

and models only the part inside the square brackets. Here j is the square root of -1. The complex vector y is a sampling of the complex signal

Note    You can also simultaneously process several signals of equal length. To do this, make x a matrix in which each signal occupies one column. The corresponding modulated signal y is a complex matrix whose kth column is the modulation of the kth column of x.

Changes for QAM

The case for quadrature amplitude modulation (QAM) is similar, except that the message signal has in-phase and quadrature components. Represent the signal using a matrix x that has an even number of columns. The odd-indexed columns represent in-phase components of the signal and the even-indexed columns represent quadrature components. If the message signal is a 2n-by-m matrix, then the modulated signal is an n-by-m matrix. As in the other methods, baseband modulation turns a real message signal into a complex modulated signal.

For example, the code below implements QAM on a set of sinusoidal input signals.

• Fs = 100; % Sampling rate is 100 samples per second.
  t = [0:1/Fs:2]'; % Sampling times
  % Signal is a four column matrix.
  % Each column models a sinusoidal signal, the frequencies
  % of which are 1 Hz, 1.5 Hz, 2 Hz, 2.5 Hz respectively.
  x = sin([2*pi*t,3*pi*t,4*pi*t,5*pi*t]);
  y = amodce(x,Fs,'qam'); % Modulate x to produce y.
  

The output below shows the sizes and types of x and y.

• whos
    Name      Size           Bytes  Class
  
    Fs        1x1                8  double array
    t       201x1             1608  double array
    x       201x4             6432  double array
    y       201x2             6432  double array (complex)
  
  Grand total is 1408 elements using 14480 bytes
  

Modulation Terminology

Simple Analog Modulation Example