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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

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

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 = 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 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

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,

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