1. Signals and Fourier Theory
Dr Costas Constantinou
School of Electronic, Electrical & Computer Engineering
University of Birmingham W: E:

2. Recommended textbook
Simon Haykin and Michael Moher, Communication Systems, 5th Edition, Wiley, 2010; ISBN:

3. Signals
A signal is a physical, measurable quantity that varies in time and/or space
Electrical signals – voltages and currents in a circuit
Acoustic signals – audio or speech signals
Video signals – Intensity and colour variations in an image
Biological signals – sequence of bases in a gene

4. Signals
In information theory, a signal is a codified message, i.e. it conveys information
We focus on a signal (e.g. a voltage) which is a function of a single independent variable (e.g. time)
Continuous-Time (CT) signals: f (t), t — continuous values
Discrete-Time (DT) signals: f [n], n — integer values only

5. Signals
Most physical signals you are likely to encounter are CT signals
Many man-made signals are DT signals
Because they can be processed easily by modern digital computers and digital signal processors (DSPs)

6. Signals Time and frequency descriptions of a signal
Signals can be represented by
either a time waveform
or a frequency spectrum

7. Fourier series
Jean Baptiste Joseph Fourier (1768 – 1830) was a French mathematician and physicist who initiated the investigation of Fourier series and their application to problems of heat transfer

8. Fourier series
A piecewise continuous periodic signal can be represented as
It follows that
Fourier showed how to represent any periodic function in terms of simple periodic functions
Thus,
where an and bn are real constants called the coefficients of the above trigonometric series

9. Fourier series
The coefficients are given by the Euler formulae

10. Fourier series
The Euler formulae arise due to the orthogonality properties of simple harmonic functions:

11. Fourier series Even and odd functions Even functions,
Thus, even functions have a Fourier cosine series
Odd functions,
Thus odd functions have a Fourier sine series

12. Fourier series Square wave, T = 1
This is an odd function, so an = 0 – we confirm this below

13. Fourier series
Similarly,

14. Fourier series
Gibbs phenomenon: the Fourier series of a piecewise continuously differentiable periodic function exhibits an overshoot at a jump discontinuity that does not die out, but approaches a finite value in the limit of an infinite number of series terms (here approx. 9%)

15. Fourier series
Pareseval’s theorem relates the energy contained in a periodic function (its mean square value) to its Fourier coefficients
Complex form: since,
we can write the Fourier series in a much more compact form using complex exponential notation

16. Fourier series It can be shown that
In the limit T → ∞, we have non-periodic signals, the sum becomes an integral and the complex Fourier coefficient becomes a function of , to yield a Fourier transform

17. Fourier transform
A non-periodic deterministic signal satisfying Dirichlet’s conditions possesses a Fourier transform
The function f (t) is single-valued, with a finite number of maxima and minima in any finite time interval
The function f (t) has a finite number of discontinuities in any finite time interval
The function f (t) is absolutely integrable
The last conditions is met by all finite energy signals

18. Fourier transform
The Fourier transform of a function is given by (here  = 2p f ),
The inverse Fourier transform is,

19. FT of a rectangular pulse
A unit rectangular pulse function is defined as
A rectangular pulse of amplitude A and duration T is thus,
The Fourier transform is trivial to compute

20. FT of a rectangular pulse
We define the unit sinc function as,
Giving us the Fourier transform pair,

21. FT of a rectangular pulse

22. FT of an exponential pulse
A decaying exponential pulse is defined using the unit step function,
A decaying exponential pulse is then expressed as,
Its Fourier transform is then,

23. Properties of the Fourier transform
Linearity
Time scaling
Duality
Time shifting
Frequency shifting

24. Properties of the Fourier transform
Area under g(t)
Area under G(t)
Differentiation in the time domain
Integration in the time domain
Conjugate functions

25. Properties of the Fourier transform
Multiplication in the time domain
Convolution in the time domain
Rayleigh’s energy theorem

26. FT of a Gaussian pulse
A Gaussian pulse of amplitude A and 1/e half-width of T is,
Its Fourier transform is given by,
In the special case

27. Signal bandwidth
Bandwidth is a measure of the extent of significant spectral content of the signal for positive frequencies
A number of definitions:
3 dB bandwidth is the frequency range over which the amplitude spectrum falls to 1/√2 = of its peak value
Null-to-null bandwidth is the frequency separation of the first two nulls around the peak of the amplitude spectrum (assumes symmetric main lobe)
Root-mean-square bandwidth

28. Signal bandwidth

29. Time-bandwidth product
For each family of pulse signals (e.g. Rectangular, exponential, or Gaussian pulse) that differ in time scale,
(duration)∙(bandwidth) = constant
The value of the constant is specific to each family of pulse signals
If we define the r.m.s. duration of a signal by,
it can be shown that,
with the equality sign satisfied for a Gaussian pulse

30. Dirac delta function
The Dirac delta function is a generalised function defined as having zero amplitude everywhere, except at t = 0 where it is infinitely large in such a way that it contains a unit area under its curve
Thus,
By definition, its Fourier transform is,

31. Spectrum of a sine wave
Applying the duality property (#3) of the Fourier transform,
In an expanded form this becomes,
The Dirac delta function is by definition real-valued and even,
Applying the frequency shifting property (#5) yields,
Using the Euler formulae that express the sine and cosine waves in terms of complex exponentials, gives,

32. Spectrum of a sine wave