Signals, Instruments, and Systems – W4An Introduction to Signal

Processing – Signals, Series, Transforms

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),( yxfI =

Signal – Definition“A signal is any time-varying or spatial-varying

quantity”

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Time

Sig

nal A

mpl

itude

Tem

pera

ture

[°C

]

Latitude [m]

Hei

ght [

m]

Lo

ngitu

de [m

]

)(tfT = )(xfh = ),( yxfh =

x [Pixel]

y [P

ixel

]

Photo: Maurizio Polese

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Continuous versus Discrete

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Continuous Discrete

1 2 3 4 5 6x

Ν∈x1 2 3 4 5 6x

R∈x2,478956983748… 4

Analog versus Digital

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Analog versus Digital

• An analog signal …– is continuous in time – has continuous values

• A digital signal …– is discrete in time – has discrete values

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• Digital– Discrete– Digital world

Analog – Digital

• Analog– Continuous– Real world

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Sensors ClassificationContinuous – Discrete

Continuous DiscreteTimeSpace

Amplitude

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Continuous: Amplitude and Time

Thermometer

Barometer

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Continuous: Time and Amplitude

BarographThermograph

Gramophone record

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Continuous: Space and AmplitudeDiscrete: Time

Camera Reel camera11

Discrete: Time and Amplitude

Digital WatchWeather station 12

Discrete: Time, Space and Amplitude

Digital camera Digital video camera13

Analog-Digital-Analog Conversion

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Analog to Digital Conversion (ADC)

Digital to Analog Conversion (DAC)

y=x(t) y=x[n]14

Analog-Digital Conversion

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Analog-Digital Converter (ADC)

• Transforms continuous analog signal into series of values

• Two key elements– Sampling (in time)– Quantization (of values)

• Two types of converters– Linear ADCs– Non-linear ADCs

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Examples of ADC Applications

• Input line of a soundcard• Sensor of digital camera• Mobile phone• Computer mouse• …

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An Example from your Course Tools

- From datasheet dsPIC on e-puck: ADC 16 channels, 12 bits, 200 ksps- 212 -1 discrete levels of resolution per channel- Max 200k Hz sampling frequency on 1 channel -> e.g., 10 channels: 20k Hz

Digital-Analog Conversion

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• Transforms a series of values into a continuous signal

• DAC outputs piecewiseconstant signal

• Additional filter stage tosmooth signal (often carriedout by the properties of the transducer/actuator itself) Time

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y[n]=0 0 -2 -4 -2 0 4 8 10 10

Digital-Analog Converter (DAC)

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Examples of DAC Applications

• Mobile phone• MP3 player• Graphics card (with older monitors)• Monitors (newer systems)

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Typical Waves and Signals

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Waves“A wave is a disturbance that propagates through space and time, usually with transference of energy.”

Wave function:

( )

fv2

2sin),(

=

=

=

−=

λ

πωλπ

ωγ

f

k

tkxtxf

“wave number”

“angular frequency”

“wave length”v = speed in x direction

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Electromagnetic Spectrum

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Common Frequencies

• Car motor: ~50 Hz (3000 rpm)• Power lines: 50 Hz• Ear: 20 Hz – 20 kHz• Ultrasound: 20 kHz – 200 MHz• Watch quartz: 32 kHz• Medical Sonograph: ~2-18 MHz• CPU: 2-3 GHz• GSM: 0.9/1.8 GHz

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Standard waveformsSine

Square

Triangle

Sawtooth

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Sinusoid (Sine Wave) in Time

)sin()( θω +⋅= tAty

phase :frequencyangular :

amplitude :

θωA

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Dirac Delta Function

∫∞+

∞−

=

≠=∞+

=

1)(

0,00,

)(

dtt

tt

t

δ

δ

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0.2

0.4

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0.8

1

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Fourier Series

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Fourier Series

• Joseph Fourier (1768-1830) proposed that any periodic function can be decomposed into a sum of simple oscillating functions, namely sines and cosines.

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“Any periodic function …”

from: Robert W. Stewart, Daniel García-Alís, “Concise DSP Tutorial” 31

“…can be decomposed into a sum of sines and cosines”from: Robert W. Stewart, Daniel García-Alís, “Concise DSP Tutorial”

∑∑ ∞

=

∞

=

+

=

10

2sin2cos)(n nn n T

ntBTntAtx ππ

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• Suppose a 2π periodic function f(t) integrable on [−π, π]

• The Fourier coefficients of x are:

1,)sin()(1

0,)cos()(1

≥=

≥=

∫

∫

−

−

ndtnttfB

ndtnttfA

n

n

π

π

π

π

π

π

∑∑ ∞

=

∞

=

+

=

10

2sin2cos)(n nn n T

ntBTntAtf ππ

Fourier Series

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Complex Numbers Review

ϕ

ϕϕ

ϕϕϕ

ϕ

sin)Im(

cos)Re(:Phase

:Magnitude

:formPolar sincos :formula sEuler'

)Im()Re( :formr Rectangula

nn

nn

n

inn

innn

CC

CC

C

eCCie

CiCC

=

=

=

+=

+=

nC

nC∠

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From Real to Complex Coefficients

∑∞

=

−−

−+

++=

10 22)(

0000

n

tintin

n

tintin

n ieeBeeAAtf

ωωωω

Real Fourier coefficients:

Using the Euler relationships:

we can express sin and cos usingexponential functions

And substituting sin and coswe get (ω0= 2π/T ):

and then

We can now express our Fourier series with complex coefficients

R,;2sin2cos)(10 ∈

+

+= ∑∞

= nnn nn BATntB

TntAAtf ππ

ieeee

iieie

iiii

i

i

2sin;

2cos

sincos)sin()cos(sincos

ωωωω

ω

ω

ωω

ωωωω

ωω

−−

−

−=

+=⇒

−=−+−=

+=

∑ ∑∞

=

−

−∞=

+

+

−

+=1

10

00

22)(

n ntinnntinnn eiBAeiBAAtf ωω

C;)( 0 ∈= ∑∞

−∞= nntin

n CeCtf ω

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Complex Fourier Coefficients

∫

∑

−

−

∞

−∞=

=

=

π

π

ω

ω

πdtetfC

eCtf

tinn

n

tinn

0

0

)(21tscoefficienFourier

)(seriesFourier

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[Matlab demo]

Examples

http://users.ece.gatech.edu/mcclella/matlabGUIs/37

Fourier Transform

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From Fourier Series to Transform• f(t)

- an aperiodic signal- view it as the limit of a periodic signal as T→ ∞- is an integrable or even a square-integrable function

• For a periodic signal, the harmonic components are spaced ω0= 2π/Tapart

• As T→ ∞, ω0→0, and harmonic components are spaced closer and closer in frequency forming a continuous spectrum

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Fourier Transform

2

2

( ) ( )

( ) ( )

i t

i t

f f t e dt

f t f e d

π ξ

π ξ

ξ

ξ ξ

∞∧−

−∞

∞ ∧

−∞

= ⋅

= ⋅

∫

∫

Fourier Transform

Inverse FourierTransform

Notation notes: ••Expression in angular frequency ω = 2πξ are also used (see tables at the end)

• In EE i is substituted by j (“i” booked for current)• transform also called “analysis equation”; inverse transform “synthesis equation”

)F(by replacedoften is )(f̂ ⋅⋅

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Common Fourier Transform

)()( ttf δ= 1)( =∧

ξf

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Frequency [Hz]

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FT of a Dirac Impulse

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From Prof. A. S. Wilsky, Signals and Systems course, to be replaced with our notation

Common Fourier Transform

1)( =tf )()( ξδξ =∧

f

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2

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0

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Frequency [Hz]

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Common Fourier Transform

)2cos()( tatf π= ( ) ( )2

)( aaf ++−=

∧ ξδξδξ

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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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0.2

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1

Frequency [Hz]Time [s]

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Common Fourier Transform

)()( trecttf = )(sinc)( ξξ =∧

f

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1

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-0.2

0

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Frequency [Hz]Time [s]

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Properties

)()()())(()(nConvolutio

)(1)()()(Scaling

)()()()(Modulation

)()()()(nTranslatio

)()()(bg(x)af(x)h(x)Linearity

02

20

0

0

ξξξ

ξξ

ξξξ

ξξ

ξξξ

ξπ

ξπ

∧∧∧

∧∧

∧∧−

∧−

∧

∧∧∧

⋅=⇒∗=

=⇒=

−=⇒=

=⇒−=

+=⇒+=

gfhxgfxh

af

ahaxfxh

fhxfexh

fehxxfxh

gbfah

ix

ix

See next lecture46

Discrete Fourier Transform

• Input and output sequences are both finite (the integral becomes a finite sum) • Matlab, operating on a computer (i.e. a digital device) can only emulate

continuity/infinity and therefore use this discrete version with an adjustable discretization level (in time, frequency, and amplitude) and finite bounds

1,,0,][X[k]

)()()(ˆ

1

0

2

2

−=⋅=

⋅==

∑

∫

−

=

−

∞

∞−

−

Nkenx

dtetfFf

N

n

knN

i

ti

π

ξπξξ

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Discrete Fourier Transform

• Different from the Discrete-Time Fourier Transform (DTFT)• Efficient implementation is done using Fast Fourier Algorithms (FFT)• Clear correspondence with the Fourier series (see slide 36)

1,,0,][1x[n]

1,,0,][X[k]

1

0

2

1

0

2

−=⋅=

−=⋅=

∑

∑

−

=

−

=

−

NkekXN

Nkenx

N

k

knN

i

N

n

knN

i

π

π DFT

Inverse DFT

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Fourier Tables

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Fourier Transforms Table

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Fourier Properties Table

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Simple Example

)()( ttriantf = )(sinc)( 2 ξξ =∧

f

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

Time [s]

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Combined Example

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Various Transforms

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Motivation for More Transforms

From Prof. A. S. Wilsky, Signals and Systems course, to be re-elaborated 55

Transforms for Causal Systems

From Prof. A. S. Wilsky, Signals and Systems course, to be re-elaborated 56

Laplace Transform

{ }

ωσ is

dttfef(t)sF st

+=

== ∫∞

−

− )()(0

L

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Fourier - Laplace

dttfesF

tftfFF

tiis

is

)(|)(

|)}({)}({)(

∫∞

∞−

−=

=

==

==

ωω

ωω L

The Fourier Transform is a special case of Laplace TransformFourier: frequency response (especially in signal processing)Laplace: impulse response (especially in control)

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Discrete-Time Fourier Transform

• Corresponds to the Fourier Transform for discrete-time signals (different from the Discrete Fourier Transform, a finite, bounded approximation of the Fourier Transform for digital devices)

• Transform discrete-time signals from time-domain to frequency domain

∑∞

−∞=

−⋅=n

nienx ωω ][]X[

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Z Transform• Corresponds to Laplace transform for time-

discrete signals• Transform signals from time-domain to frequency

domain• The Discrete-Time Fourier Transform is a special

case of the Z Transform

)sin(cosor

][]}[{)(

ϕϕϕ jAzAez

znxnxzX

jn

n

+==

== ∑∞

−∞=

−Z

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Transform Overview

Discrete signalTime domain

Discrete signalFrequency domain

Continuous signalFrequency domain

Continuous signalTime domain

Fourier/Laplace

Inverse Fourier/Laplace

Z/DTFT

Inverse Z/DTFT

DA

C

AD

C

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Conclusion

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Take Home Messages• A signal can be a varying quantity in time and/or space• Signal classification: continuous vs. discrete in time,

space and amplitude• Fourier decomposition: every periodic signal can be

decomposed into a sum of sines and cosines• Fourier coefficients are the weights in this sum• Fourier transform for aperiodic signals• Causal, potentially unstable systems requires Laplace

transform (in continuous time) and Z transform (in discrete time) to analyze their response in frequency domain

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Additional Literature – Week 7Book• McClellan J. , Schafer R., Yoder M. “DSP First: A

Multimedia Approach”, Prentice Hall, 1998

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