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# Signals and Fourier Theory - PowerPoint PPT Presentation

Signals and Fourier Theory. Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W: www.eee.bham.ac.uk/ConstantinouCC/ E: [email protected] Recommended textbook.

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### Signals and Fourier Theory

Dr Costas Constantinou

School of Electronic, Electrical & Computer Engineering

University of Birmingham

W: www.eee.bham.ac.uk/ConstantinouCC/

E: [email protected]

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

• 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

• 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

• 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)

• Time and frequency descriptions of a signal

• Signals can be represented by

• either a time waveform

• or a frequency spectrum

• 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

• 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

• The coefficients are given by the Euler formulae

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

• Even and odd functions

• Even functions,

• Thus, even functions have a Fourier cosine series

• Odd functions,

• Thus odd functions have a Fourier sine series

• Square wave, T = 1

• This is an odd function, so an = 0 – we confirm this below

• Similarly,

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%)

• 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

• 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

• 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

• The Fourier transform of a function is given by (here  = 2pf),

• The inverse Fourier transform is,

• 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

• We define the unit sinc function as,

• Giving us the Fourier transform pair,

• A decaying exponential pulse is defined using the unit step function,

• A decaying exponential pulse is then expressed as,

• Its Fourier transform is then,

• Linearity

• Time scaling

• Duality

• Time shifting

• Frequency shifting

• Area under g(t)

• Area under G(t)

• Differentiation in the time domain

• Integration in the time domain

• Conjugate functions

• Multiplication in the time domain

• Convolution in the time domain

• Rayleigh’s energy theorem

• A Gaussian pulse of amplitude A and 1/e half-width of T is,

• Its Fourier transform is given by,

• In the special case

• 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

• 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

• 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,

• 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,