Signals, Instruments, and Systems – W5

Introduction to Signal Processing – Convolution, Sampling, Reconstruction

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Motivation from Week 1 Lecture

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Sensing

Processing Communication

Processing

Mobility

In-situ instrument

Visualization

Storing

Transportation channel Base station

Highlighted blocks are those mainly leveraging the content of this lecture.

Convolution

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Convolution

dtgftgf )()()(

• For each value of t:1. Flip (reflect) g2. Shift g by t3. Multiply f and g4. Integrate over

• Note that the result does not depend on !

dtgf

tgftgg

gg

)()( 4)

)()( 3))()( 2)

)()( 1)

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[Matlab demo “cconvdemo”]

http://users.ece.gatech.edu/mcclella/matlabGUIs/5

Examples

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Examples

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Examples

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Examples

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Examples

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Convolution in Time and Frequency Domains

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))(ˆˆ()(ˆ)()()(

)(ˆ)(ˆ)(ˆ))(()(

gfhtgtfth

gfhtgfth

Time domain Frequency domain

Ord

inar

yfr

eque

ncy

))((21)()()()(

)()()())(()(

GFHtgtfth

GFHtgfth

Ang

ular

freq

uenc

y

Convolution Properties

)()()(

)()()(

)()(

agfgafgfa

hfgfhgf

hgfhgf

fggf

tionmultiplica scalar ity withAssociativ

vityDistributi

ityAssociativ

ityCommutativ

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Discrete Convolution

m

mngmfngf

dtgftgf

][][][

)()()(

Notes:• Similar to the continuous version• The integral becomes an infinite sum• Matlab, operating on a computer (i.e. a digital device) can only emulate

continuity and therefore use the discrete version with an adjustable discretization level (in time and amplitude) and finite bounds of the convolution window 13

[Matlab demo “dconvdemo”]

http://users.ece.gatech.edu/mcclella/matlabGUIs/14

Sampling

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• Digital– Discrete– Digital world

Analog – Digital

• Analog– Continuous– Real world

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

4

6

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Time

Sig

nal A

mpl

itude

Time0 100 200 300 400 500 600 700 800 900 1000

-4

-2

0

2

4

6

8

10

Sig

nal A

mpl

itude

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Train of Dirac Impulses

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n

nTttx )()(

Train of Dirac Impulses

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n

T

T

tjk

n

Tk

TX

Tdtetx

Tatx

nTttx

22)(

1)(1)(

)()(

2/

2/

0

Time domain Frequency domain

F()

Sampling in Time Domain

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nnp

n

nTtnTxnTttxtptxtx

nTttp

tx

)()()()()()()(

)()(

)(

Train of Dirac pulses

T = sampling periodf = 1/T = sampling

frequency

Sampling in Frequency Domain

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kS

kSp

kS

p

p

kXT

kXT

X

kT

P

PXX

tptxtx

)(1

)(*)(1)(

)(2)(

)(*)(21)(

)()()(

Ts 2

Samplingfrequency

Convolution propertyTime domain

Frequency domain

Sampling a Band-Limited Signal

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kS

p

kXT

PXX

)(1

)(*)(21)(

k

SkT

P )(2)(

X(ω) → spectrum of signal x(t) with highest frequency < ωm

Sampling frequency ωs > 2 ωm

Sampling a Band-Limited Signal

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X(ξ) → spectrum of signal x(t) with highest frequency < 2 kHz

Sampling frequency: 5 kHz > 2 * 2 kHz

Sampling a Band-Limited Signal

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X(ξ) → spectrum of signal x(t) with highest frequency < 3 kHz

Sampling frequency: 5 kHz < 2 * 3 kHz

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Original Signal

)52sin(2.0)22sin(4.0)2sin()( ttttf 24

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Too Few Samples (1Hz)

→ Data is lost25

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Too Many Samples (100 Hz)

→Redundant data→Increase of data size

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Minimal Possible Sampling (> 10 Hz)

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Nyquist–Shannon Theorem• If a function x(t) contains no frequencies higher than

B Hz, it is completely determined by giving its coordinates at a series of points spaced 1/(2B) seconds apart.

• Sampling frequency must be at least two times greater than the maximal signal frequency

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Sampling in Practice

• Sampling frequency two times greater than maximal frequency is the limit

• If possible, try to use a sampling frequency 10 times greater than the maximal frequency

• Audio CD, sampling at 44.1 kHzMaximal hearable frequency: 20 kHz

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Signal Reconstruction

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spectrum of original signal

spectrum of reconstructedsignal

)(tx x

n

nTttp )(

)(txr)(H

spectrum of sampled signalsampling frequency ωs > 2 ωm

filtering filter cut-off frequency ωm < ωc < (ωs –ωm)

If there is no overlap between the shifted spectra the signal xr(t) can beperfectly reconstructed from x(t)

)(txp

)()()( HXX pr

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spectrum of original signal

spectrum of reconstructedsignal

)(tx x

n

nTttp )(

)(txr)(H

spectrum of sampled signalsampling frequency fs < 2 fm

filtering filter cut-off frequency fcfm < fc < (fs –fm)

If there is overlap between the shifted spectra the signal xr(t) cannot beperfectly reconstructed from x(t)

)(txp

)()()( HXX pr

)(tx x

n

nTttp )(

)(txr

Time Domain Interpretationof Signal Reconstruction

ωm < ωc < (ωs –ωm)

nTt

nTtTnTx

nTthnTx

thnTtnTx

thtxtx

c

n

n

n

pr

sin

)(*

)(*)()(

t

tTth c

sin)( with

F-1()

The lowpass filter interpolates the samples assuming x(t) contains no energy at frequencies > ωc(ωc =cutoff frequency)

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34From Prof. A. S. Willsky, Signals and Systems course

Signal Reconstruction Methods

snss

s

s

ns

TtnTtnTxtx

TnTtnTxtx

sinc)()()(

sinc)()(

• Signal has to be band limited (i.e. Fourier transform for frequencies greater than B equal 0)

• The sampling rate must exceed twice the bandwidth, 2B

BTs 2

1

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Bfs 2or

1. Zero-order hold2. First-order hold – linear interpolation 3. Whittaker-Shannon interpolation (bandlimited interpolation):

(Alternative equivalent formulation)

36From Prof. A. S. Willsky, Signals and Systems course

Aliasing

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No Problems in Reconstruction

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Reconstruction Problems

Alias

Overlapping

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Harmonics

Fundamental Frequency

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1Hz2Hz3Hz4Hz

Harmonics

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La-Tone (440 Hz) sampled at 44.1 kHz (CD standard)

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La-Tone (440 Hz) sampled at 2 kHz without filtering

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La-Tone (440Hz) sampled at 2 kHz filtered at 1 kHz

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Aliasing Audio Examples

Original sound Aliases 4 kHz Correct sampling 4 kHz

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Moiré Pattern

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Aliasing Video Examples

http://www.youtube.com/watch?v=jHS9JGkEOmA

https://www.youtube.com/watch?v=R-IVw8OKjvQ

Conclusion

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Take Home Messages

When you sample a signal …

1. Make sure you know what the maximum frequency fmax is or enforce it through a low-pass filter

2. Make sure you sample at fs > 2 fmax(Nyquist rule)

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Books:• Ronald W. Schafer and James H. McClellan

“DSP First: A Multimedia Approach”, 1998• A. Oppenheim and A. S. Willsky with S. Hamid,

“Signals and Systems”, Prentice Hall, 1996.

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Additional Literature – Week 5