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

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Signals and Fourier Theory

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Signals and Fourier Theory

Dr Costas Constantinou

School of Electronic, Electrical & Computer Engineering

University of Birmingham



Recommended textbook

  • 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

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  = 2pf),

  • 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

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