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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: c.constantinou@bham.ac.uk

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

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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

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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

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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)

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

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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

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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 a n and b n are real constants called the coefficients of the above trigonometric series

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Fourier series The coefficients are given by the Euler formulae

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

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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

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

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Fourier series Similarly,

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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%)

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Fourier series Paresevals 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

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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

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Fourier transform A non-periodic deterministic signal satisfying Dirichlets conditions possesses a Fourier transform 1.The function f (t) is single-valued, with a finite number of maxima and minima in any finite time interval 2.The function f (t) has a finite number of discontinuities in any finite time interval 3.The function f (t) is absolutely integrable – The last conditions is met by all finite energy signals

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

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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

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

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FT of a rectangular pulse

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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,

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Properties of the Fourier transform 1.Linearity 2.Time scaling 3.Duality 4.Time shifting 5.Frequency shifting

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Properties of the Fourier transform 6.Area under g(t) 7.Area under G(t) 8.Differentiation in the time domain 9.Integration in the time domain 10.Conjugate functions

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Properties of the Fourier transform 11.Multiplication in the time domain 12.Convolution in the time domain 13.Rayleighs energy theorem

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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

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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

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Signal bandwidth

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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

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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,

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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,

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Spectrum of a sine wave

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