I. Definition

1. What is DSP?

DSP is for Digital Signal Processing. It can be defined as the mathematical manipulation of analogue signals that have been converted to their digital form.

A signal is an analogue quantity that conveys and represents information of some form (acoustic signal, radio…). It can be converted to a digital form (to binary numerical values) via the Fourier theorem.


2. Theoretical background

Why Fourier? Because we want to convert the signal from the time domain to the frequency domain, which allows a much more easy and powerful way to signal processing than the time domain.

If x(t) is the equation of the signal, then its Fourier transform equation X(w) is:

X(w) = ò x(t).ej(-wt) dt

This equation has a similar reverse form, which can be used to transfer the signal from the frequency domain back to the time domain:


x(t) = ò X(w).ej(+wt) dt


Examples:

· Sinusoidal signal


f(t) = A.cos(w0.t)

Its Fourier transform equation is:

F(w) = p.A.[ d(w - w0 ) + d( w + w-0 )]

( with d the Dirac function)


· Impulsion (digital pulse):


w(t) = 1 if t belongs to [-t/2, t/2]
w(t) = 0 elsewhere





Then its Fourier transform equation is :

W(w) = t.sinc(w.t/2)





· Series of impulsions:

Let's take a series of impulsions:


Its Fourier transform equation will have the following shape:



Any signal in the time domain has its equivalent in the frequency domain. It is now possible to manipulate the Fourier transformed signal.



Examples:

a) Distortion of a sine function

Let's take a pure sine signal, with its Fourier transform function.






Let's put now a distorsion in the signal. If we visualise the frequency domain of the distorded signal, we will obtain:





If we want to avoid the distortion, we can put the signal through a low-pass filter, in order to eliminate the high frequencies. The signal has been removed from its distortions.


b) Noise in an acoustic signal

By the same way, it is possible in an acoustic signal to reduce the high frequency hiss. We have manipulated the signal via its Fourier transform.



3. More information


The following websites and documents will give you more details on the theory of DSP:

· Blue Water Systems (http://www.bluews.com/products/training/tutor)

BlueWS is a manufacturer of DSP boards and subsystems. They have also a good theoretical introduction on DSPs.


· University of Strathclyde (http://www.spd.eee.strath.ac.uk/~interact/FFT/fourier.html)

A basic tutorial on the Fourier function and Signal Processing, with several examples.


· A document in .pfd format is also available under the following path:

L:\PROJECTS\ERTMS\TRAINBRN\DSP
This document, from the University of Strathclyde, is under the form of a lecture given to undergraduate students (good start).