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

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

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

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

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)

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

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

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

Fourier series The coefficients are given by the Euler formulae

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

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

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

Fourier series Similarly,

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%)

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

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

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

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

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

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

FT of a rectangular pulse

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,

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

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

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

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

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

Signal bandwidth

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

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,

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,

Spectrum of a sine wave