Review on Fourier … - PowerPoint PPT Presentation

Review on Fourier …

• Prof. Brian L. Evans and Mr. Dogu Arifler Dept. of Electrical and Computer Engineering The University of Texas at Austin course:

EE 313 Linear Systems and Signals Fall 2003

Fourier Series

• Sound clips
• Sinusoid with frequency of 660 Hz (no harmonics)
• Square wave with fundamental frequency of 660 Hz
• Sawtooth wave with fundamental frequency of 660 Hz
• Beat frequencies at 660 Hz +/- 12 Hz
• Spectrogram representation
• Time on the horizontal axis
• Frequency on the vertical axis

Transmitter

Receiver

Channel

Receiver

Transmitter

upstream

Home

ServiceProvider

downstream

• FM radio
• Single carrier at radio station frequency (e.g. 94.7 MHz)
• Bandwidth of 165 kHz: left audio channel, left – right audio channels, pilot tone, and 1200 baud modem
• Station spacing of 200 kHz
• Modulator/Demodulator (Modem)

Courtesy of Milos Milosevic (Schlumberger)

Courtesy of Shawn McCaslin (Cicada Semiconductor, Austin, TX)

• Discrete Multitone (DMT) modulation

ADSL (ANSI 1.413) and proposed for VDSL

• Orthogonal Freq. Division Multiplexing (OFDM)

Digital audio/video broadcasting (ETSI DAB-T/DVB-T)

Courtesy of Güner Arslan (Cicada Semiconductor)

channel frequency response

magnitude

a carrier

a subchannel

frequency

Harmonically related carriers

• f(t) is periodic if, for some positive constant T0

For all values of t, f(t) = f(t + T0)

Smallest value of T0 is the period of f(t).

sin(2pfot) = sin(2pf0t + 2p) = sin(2pf0t + 4p): period 2p.

• A periodic signal f(t)

Unchanged when time-shifted by one period

Two-sided: extent is t (-, )

May be generated by periodically extending one period

Area under f(t) over any interval of duration equal to the period is the same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

• f0(t) = C0 cos(2 p f0 t + q0)
• fn(t) = Cn cos(2 p n f0 t + qn)
• The frequency, n f0, is the nth harmonic of f0
• Fundamental frequency in Hertz is f0
• Fundamental frequency in rad/s is w = 2 p f0

Cn cos(n w0 t + qn) =Cn cos(qn) cos(n w0 t) - Cn sin(qn) sin(n w0 t) =an cos(n w0 t) + bn sin(n w0 t)

• General representationof a periodic signal
• Fourier seriescoefficients
• Compact Fourierseries

• Existence
• Convergence for all t
• Finite number of maxima and minima in one period of f(t)

T0 = p

Fundamental frequency

f0 = 1/T0 = 1/p Hz

w0 = 2p/T0 = 2 rad/s

f(t)

1

e-t/2

-p

0

p

T0 = 2

Fundamental frequency

f0 = 1/T0 = 1/2 Hz

w0 = 2p/T0 = p rad/s

f(t)

A

-1

0

1

-A

T0 = 2p

Fundamental frequency

f0 = 1/T0 = 1/2p Hz

w0 = 2p/T0 = 1 rad/s

f(t)

1

-p/2

-p

-2p

p/2

p

2p

Fourier Analysis

• For all t,x(t + T) = x(t)

x(t) is a period signal

• Periodic signals havea Fourier seriesrepresentation
• Cncomputes the projection (components) of x(t) having a frequency that is a multiple of the fundamental frequency 1/T.

• Conditions for the Fourier transform of g(t) to exist (Dirichlet conditions):

x(t) is single-valued with finite maxima and minima in any finite time interval

x(t) is piecewise continuous; i.e., it has a finite number of discontinuities in any finite time interval

x(t) is absolutely integrable

• Generalized frequency variable s = s + j w
• Laplace transform consists of an algebraic expression and a region of convergence (ROC)
• For the substitution s = j w or s = j 2 p f to be valid, the ROC must contain the imaginary axis

• What system properties does it possess?
• Memoryless
• Causal
• Linear
• Time-invariant
• What does it tell you about a signal?
• Answer: Measures frequency content
• What doesn’t it tell you about a signal?
• Answer: When those frequencies occurred in time

• Unit gate function (a.k.a. unit pulse function)
• What does rect(x / a) look like?
• Unit triangle function

rect(x)

1

x

-1/2

0

1/2

D(x)

1

x

-1/2

0

1/2

sinc(x)

1

x

-3p

-2p

-p

0

p

2p

3p

• Sinc function
• Even function
• Zero crossings at
• Amplitude decreases proportionally to 1/x

F(w)

f(t)

t

F

1

w

t

-t/2

0

t/2

-6p

-4p

-2p

2p

4p

6p

0

t

t

t

t

t

t

f(t) = 1

1

t

0

F

F(w) = 2 p d(w)

(2p)

w

0

(2p) means that the area under the spike is (2p)

F

F(w)

f(t)

(p)

(p)

t

w

0

-w0

w0

0

t

sgn(t)

1

-1

Fourier Transform Properties

Assuming that Re{a} > 0

t

F(w)

w

-6p

-4p

-2p

2p

4p

6p

t

t

t

t

t

t

0

• Forward/inverse transforms are similar
• Example: rect(t/t)  t sinc(wt / 2)
• Apply duality t sinc(t t/2)  2 p rect(-w/t)
• rect(·) is even t sinc(t t /2)  2 p rect(w/t)

f(t)

1

t

-t/2

0

t/2

• Same as Laplacetransform scaling property

|a| > 1: compress time axis, expand frequency axis

|a| < 1: expand time axis, compress frequency axis

• Effective extent in the time domain is inversely proportional to extent in the frequency domain (a.k.a bandwidth).

f(t) is wider  spectrum is narrower

f(t) is narrower  spectrum is wider

• Shift in time
• Does not change magnitude of the Fourier transform
• Does shift the phase of the Fourier transform by -wt0 (so t0 is the slope of the linear phase)

F(w)

1

w

-w1

w1

0

• Example: y(t) = f(t) cos(w0 t)

f(t) is an ideal lowpass signal

Assume w1 << w0

• Demodulation is modulation followed by lowpass filtering
• Similar derivation for modulation with sin(w0 t)

Y(w)

1/2 F(w+w0)

1/2 F(w-w0)

1/2

w

-w0 - w1

-w0 + w1

w0 - w1

w0 + w1

0

-w0

w0

f(t)  0, when |t| 

f(t) is differentiable

Derivation of property:Given f(t) F(w)

• Definition of Fourier Transform
• Two ways to find Fourier Transform
• Use definitions
• Use properties