Electrical Communications Systems ECE.09.331 Spring 2009

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Electrical Electrical Communications SystemsCommunications Systems

ECE.09.331ECE.09.331 Spring 2009Spring 2009

Shreekanth MandayamECE Department

Rowan University

http://engineering.rowan.edu/~shreek/spring09/ecomms/

Lecture 3bLecture 3bFebruary 4, 2009February 4, 2009


PlanPlan• Recall: CFT’s (spectra) of common waveforms

• Impulse• Sinusoid• Rectangular Pulse

• CFT’s for periodic waveforms• Sampling

• Time-limited and Band-limited waveforms• Nyquist Sampling• Impulse Sampling• Dimensionality Theorem

• Discrete Fourier Transform (DFT)• Fast Fourier Transform (FFT)


ECOMMS: TopicsECOMMS: Topics

Probability

Inform ation

Entropy

Channel Capacity

Discrete

Pow er & Energy Signals

Continuous Fourier Transform

Discrete Fourier Transform

Baseband and Bandpass Signals

Com plex Envelope

Gaussian Noise & SNR

Random VariablesNoise Calculations

Continuous

Signals

AMSw itching M odulator

Envelop Detector

DSB-S CProduct M odulatorCoherent Detector

Costas Loop

SSBW eaver's MethodPhasing M ethod

Frequency M ethod

Frequency & Phase M odulationNarrowband/WidebandVCO & Slope Detector

PLL

Analog

Source EncodingHuffm an codes

Error-control EncodingHam m ing Codes

Sam plingPAM

QuantizationPCM

Line Encoding

Tim e Division M uxT1 (DS1) Standards

Packet Sw itchingEthernet

ISO 7-Layer Protocol

BasebandCODEC

ASKPSKFSK

BPSK

QPSK

M -ary PSK

QAM

BandpassM ODEM

DigitalDigital Com m Transceiver

Systems

Electrical Comm unication Systems


Recall: DefinitionsRecall: Definitions

)f(j

ft2j

e )f(W)f(W

)f(Y j)f(X)f(W

dte )t(w)t(w)f(W

F

Continuous Fourier Transform (CFT)

Frequency, [Hz]

AmplitudeSpectrum

PhaseSpectrum

dfe )f(W)f(W)t(w ft2j1-

F

Inverse Fourier Transform (IFT)

See p. 45Dirichlet Conditions


CFT’s of Common WaveformsCFT’s of Common Waveforms

• Impulse (Dirac Delta)

• Sinusoid

• Rectangular PulseMatlab Demo:recpulse.m


CFT for Periodic SignalsCFT for Periodic SignalsRecall:

dte )t(w)f(W ft2j

CFT: Aperiodic Signals

dte )t(wT1

W

where

e W)t(w

2/T

2/T

tnf2j

0n

tnf2j

nn

0

0

0

0

FS: Periodic Signals

• We want to get the CFT for a periodic signal

• What is ? tnfje 02 F


CFT for Periodic SignalsCFT for Periodic Signals

• Sine Wave

w(t) = A sin (2f0t)

• Square Wave

A

-AT0/2 T0

Instrument Demo


SamplingSampling

• Time-limited waveform

w(t) = 0; |t| > T

• Band-limited waveform

W(f)=F{(w(t)}=0; |f| > B

-T T

w(t)

t -B B

W(f)

f

• Can a waveform be both time-limited and band-limited?


Nyquist Sampling TheoremNyquist Sampling Theorem

n

ss

ss

n

fn

tf

fn

tf

atw

sin

)(

)(twfa sn

sfn

n wa

• Any physical waveform can be represented by

• where

• If w(t) is band-limited to B Hz and Bfs 2


What does this mean?What does this mean?

1/fs 2/fs 3/fs 4/fs 5/fs

w(t)

t

a3 = w(3/fs)

sf

3tsf

sf

3tsfsin

• If then we can reconstruct w(t) without error by summing weighted, delayed sinc pulses• weight = w(n/fs)• delay = n/fs

• We need to store only “samples” of w(t), i.e., w(n/fs)

• The sinc pulses can be generated as needed (How?)

Bfs 2

Matlab Demo:sampling.m


Impulse SamplingImpulse Sampling

• How do we mathematically represent a sampled waveform in the

• Time Domain?• Frequency Domain?


Sampling: Spectral EffectSampling: Spectral Effect

w(t)

t

ws(t)

t

f-B 0 B

|W(f)|

f

|Ws(f)|

-2fs -fs 0 fs 2 fs

(-fs-B) -(fs +B) -B B (fs -B) (fs +B)

F

F

Original

Sampled


Spectral Effect of SamplingSpectral Effect of Sampling

Spectrum of a

“sampled” waveform

Spectrum of the

“original” waveform replicated every fs Hz

=


AliasingAliasing

• If fs < 2B, the waveform is “undersampled”

• “aliasing” or “spectral folding”

• How can we avoid aliasing?

• Increase fs

• “Pre-filter” the signal so that it is bandlimited to 2B < fs


Dimensionality TheoremDimensionality Theorem• A real waveform can be completely specified by

N = 2BT0 independent pieces of information over a time interval T0

• N: Dimension of the waveform• B: Bandwidth

• BT0: Time-Bandwidth Product

• Memory calculation for storing the waveform• fs >= 2B

• At least N numbers must be stored over the time interval T0 = n/fs


Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT)• Discrete Domains

• Discrete Time: k = 0, 1, 2, 3, …………, N-1• Discrete Frequency: n = 0, 1, 2, 3, …………, N-1

• Discrete Fourier Transform

• Inverse DFT

Equal time intervals

Equal frequency intervals

1N

0k

nkN2

j;e ]k[x]n[X

1N

0n

nkN2

j;e ]n[X

N1

]k[x

n = 0, 1, 2,….., N-1

k = 0, 1, 2,….., N-1


Importance of the DFTImportance of the DFT• Allows time domain / spectral domain transformations

using discrete arithmetic operations

• Computational Complexity• Raw DFT: N2 complex operations (= 2N2 real operations)• Fast Fourier Transform (FFT): N log2 N real operations

• Fast Fourier Transform (FFT)• Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the

periodicity and symmetry of e-j2kn/N

• VLSI implementations: FFT chips• Modern DSP


How to get the frequency axis in the DFTHow to get the frequency axis in the DFT

• The DFT operation just converts one set of number, x[k] into another set of numbers X[n] - there is no explicit definition of time or frequency

• How can we relate the DFT to the CFT and obtain spectral amplitudes for discrete frequencies?

1N

0

x

.

x

]k[x

1N

0

X

.

X

]n[X

(N-point FFT)

n=0 1 2 3 4 n=N

f=0 f = fs

N

fs

Need to know fs


SummarySummary

Electrical Communications Systems ECE.09.331 Spring 2009

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