Signals and Fourier Theory

Dr Costas ConstantinouSchool of Electronic, Electrical & Computer Engineering

University of BirminghamW: 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

f t T f t

f t nT f t

1, cos ,sin , cos 2 ,sin 2 , cos ,sin , , 2t t t t n t n t T

01

cos sinn nn

f t a a n t b n t

Fourier series

• The coefficients are given by the Euler formulae

2

0

2

1T

T

a f t dtT

2

2

2cos

T

n

T

a f t n t dtT

2

2

2sin

T

n

T

b f t n t dtT

Fourier series

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

0 when sin sin

when

0 when cos cos

when

sin cos 0 for all ,

n mnx mxdx

n m

n mnx mxdx

n m

nx mxdx n m

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

0, for all ng t g t b n

01

cosnn

g t a a n t

1

sinnn

h t a n t

0, for all nh t h t a n

Fourier series

• Square wave, T = 1

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

1

2

12

when 0and 1

when 0

V tf t f t f t

V t

12

12

12

12

12

12

0

0

0

0

2 cos 2

2 cos 2 cos 2

1 12 sin 2 sin 2

2 2

sin 0 sin sin sin 0 0

na f t nt dt

V nt dt V nt dt

V nt ntn n

Vn n

n

Fourier series

• Similarly,

12

12

12

12

12

12

0

0

0

0

2 sin 2

2 sin 2 sin 2

1 12 cos 2 cos 2

2 2

cos0 cos cos cos0

21 cos

4for odd

0 for even

nb f t nt dt

V nt dt V nt dt

V nt ntn n

Vn n

nV

nnV

nn

n

Fourier seriesGibbs 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

2

2 2 2 20

12

1 1 1

4 2

T

n nnT

f t dt a a bT

cos and sin2 2

jn t jn t jn t jn te e e en t n t

j

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

jn tn

n

f t c e

2

2

1T

jn tn

T

c f t e dtT

0 0

1 1, ,

2 2n n n n n nc a jb c a jb c a

Fourier transform

• A non-periodic deterministic signal satisfying Dirichlet’s conditions possesses a Fourier transform1. The function f (t) is single-valued, with a finite number of

maxima and minima in any finite time interval2. The function f (t) has a finite number of discontinuities in

any finite time interval3. The function f (t) is absolutely integrable

– The last conditions is met by all finite energy signals

g t dt

2g t dt

Fourier transform

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

• The inverse Fourier transform is,

exp 2G f g t j ft dt

exp 2g t G f j ft df

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

1 12 2

12

1,rect

0,

tt

t

rectt

g t AT

2

2

2

2

rect exp 2 exp 2

sinexp 2

2

T

T

T

T

tG f A j ft dt A j ft dt

T

fTAj ft AT

j f fT

FT of a rectangular pulse

• We define the unit sinc function as,

• Giving us the Fourier transform pair,

sinsinc

rect sinct

A AT fTT

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,

12

1, 0

, 0

0, 0

t

u t t

t

exp , 0g t u t at a

0

0

exp exp 2

exp 2

1

2

G f at j ft dt

t a j f dt

a j f

Properties of the Fourier transform

1. Linearity

2. Time scaling

3. Duality

4. Time shifting

5. Frequency shifting

1 2 1 2ag t bg t aG f bG f

1 fg at G

a a

g t G f G t g f

0 0exp 2g t t G f j ft

exp 2 c cg t j f t G f f

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

0g t dt G

2d

g t j f G fdt

01

2 2

t Gg d G f f

f

* *g t G f g t G f

0g G f df

Properties of the Fourier transform

11. Multiplication in the time domain

12. Convolution in the time domain

13. Rayleigh’s energy theorem

1 2 1 2g t g t G G f d

2 2g t dt G f df

1 2 1 2g g t d G f G f

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

2 2expg t A t T

2 2

22 2 2

2 2 2

exp exp 2

exp exp

exp

G f A t T j ft dt

tA f T A j fT dt

T

AT f T

2 21 , exp expT t f

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

1 222

2rms

f G f dfW

G f df

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

1 222

2rms

t g t dtT

g t dt

1 4rms rmsT W

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,

0, 0 and 1t t t dt

0 0g t t t dt g t

exp 2 1 1f t j ft dt t

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,

1 f

exp 2j ft dt f

cos 2 ft dt f

exp 2 c cj f t f f

Spectrum of a sine wave

1cos 2

2c c cf t f f f f

1sin 2

2c c cf t f f f fj