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Review on Fourier …. Slides edited from: 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. Spectrogram Demo ( DSP First ). Sound clips

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slide2
Slides edited from:
  • 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

spectrogram demo dsp first
Spectrogram Demo (DSP First)
  • 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
frequency content matters

Transmitter

Receiver

Channel

Receiver

Transmitter

upstream

Home

ServiceProvider

downstream

Frequency Content Matters
  • 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)
demands for broadband access
Demands for Broadband Access

Courtesy of Milos Milosevic (Schlumberger)

dsl broadband access standards
DSL Broadband Access Standards

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

multicarrier modulation
Multicarrier Modulation
  • 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

periodic signals
Periodic Signals
  • 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

sinusoids
Sinusoids
  • 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)

fourier series1
Fourier Series
  • General representationof a periodic signal
  • Fourier seriescoefficients
  • Compact Fourierseries
existence of the fourier series
Existence of the Fourier Series
  • Existence
  • Convergence for all t
  • Finite number of maxima and minima in one period of f(t)
example 1
Fundamental period

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

Example #1
example 2
Fundamental period

T0 = 2

Fundamental frequency

f0 = 1/T0 = 1/2 Hz

w0 = 2p/T0 = p rad/s

f(t)

A

-1

0

1

-A

Example #2
example 3
Fundamental period

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

Example #3
periodic signals1
Periodic Signals
  • 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.
fourier integral
Fourier Integral
  • 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

laplace transform
Laplace Transform
  • 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
fourier transform
Fourier Transform
  • 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
useful functions
Useful Functions
  • 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

useful functions1

sinc(x)

1

x

-3p

-2p

-p

0

p

2p

3p

Useful Functions
  • Sinc function
    • Even function
    • Zero crossings at
    • Amplitude decreases proportionally to 1/x
fourier transform pairs

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

Fourier Transform Pairs
fourier transform pairs1

f(t) = 1

1

t

0

F

Fourier Transform Pairs

F(w) = 2 p d(w)

(2p)

w

0

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

fourier transform pairs2

F

Fourier Transform Pairs

F(w)

f(t)

(p)

(p)

t

w

0

-w0

w0

0

duality

t

F(w)

w

-6p

-4p

-2p

2p

4p

6p

t

t

t

t

t

t

0

Duality
  • 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

scaling
Scaling
  • 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

time shifting property
Time-shifting Property
  • 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)
modulation1

F(w)

1

w

-w1

w1

0

Modulation
  • 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

time differentiation property
Conditions

f(t)  0, when |t| 

f(t) is differentiable

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

Time Differentiation Property
summary
Summary
  • Definition of Fourier Transform
  • Two ways to find Fourier Transform
    • Use definitions
    • Use properties