Signals and Fourier Theory

1. Signals and Fourier Theory Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W: www.eee.bham.ac.uk/ConstantinouCC/ E: c.constantinou@bham.ac.uk

2. Recommended textbook • Simon Haykin and Michael Moher, Communication Systems, 5th Edition, Wiley, 2010; ISBN:970-0-470-16996-4

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  = 2pf), • 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 = 0.707 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