Convolution of signals in communications

Convolution is a fundamental operation in signal processing, particularly in the field of communications. It is essentially a mathematical operation that combines two signals to produce a third signal, which represents the output of a system when the input is one of the original signals.

Why convolution?

System Modeling: Convolution is used to model the behavior of linear time-invariant (LTI) systems. These systems are characterized by their impulse response, which is essentially the output of the system when a very short impulse is applied as input.

Filtering: Convolution is the basis for filtering operations, which are used to remove unwanted frequencies or noise from a signal.

Modulation: Convolution is involved in various modulation techniques, such as pulse shaping in QAM.

The Convolution Integral

ystem Modeling: Convolution is used to model the behavior of linear time-invariant (LTI) systems. These systems are characterized by their impulse response, which is essentially the output of the system when a very short impulse is applied as input.
Filtering: Convolution is the basis for filtering operations, which are used to remove unwanted frequencies or noise from a signal.
Modulation: Convolution is involved in various modulation techniques, such as amplitude modulation (AM) and frequency modulation (FM).

Mathematically, the convolution of two continuous-time signals x(t) and h(t) is defined as:

$y(t) = x(t) * h(t) = \int\limits_{ - \infty }^{ + \infty } {x(\tau ) \cdot h(t - \tau )d\tau }$

where:

y(t) is the output signal - convolved signal.
x(t) is the input signal - signal 1.
h(t) is the impulse response of the system - signal 2.
τ is a dummy variable of integration.

y(t) is the output signal.
x(t) is the input signal.
h(t) is the impulse response of the system.
τ is a dummy variable of integration.

Graphical Interpretation

Convolution can be visualized graphically as the flipping and sliding of one signal over another. The output at a particular time is the integral of the product of the two signals at that time.

This experiment

In this experiment we convolute different single-period signals.

$x (t)$	$h (t)$	$y (t)$
$x (t)$	$h (t)$	$y (t)$
$x (t)$	$h (t)$	$y (t)$
$x (t)$	$h (t)$	$y (t)$

First row

In the first row we can see a triangle signal. To convolve a triangle signal we took 2 identical rectangular signals.

What stands out if you compare the length of the signals?

Second row

In the second row we combine a longer rectangle with a shorter rectangle. The convolved signal is a trapeze.

What stands out if you compare the length of the signals?

Third row

In the third row we combine a triangle with another rectangle.

What stands out if you compare the length of the signals?

Which signal is similar to the convolved signal?

Fourth row

In the last row we convolve two raised cosine signals. This is a typical Nyquist signal, where the zero-crossing is in a predetermined interval.

How long is the zero-crossing of the convoluted signal in comparison with the first two signals?

Related experiments, images or signals:

Convolution of two raised cosine signals

Convolution of two rectanglar pulses (1)

Convolution of two Gauss