introduction to signal processing - disalw3.epfl.ch

1

Signals, Instruments, and Systems – W4

Introduction to Signal Processing – System

Properties, Responses, Functions, More Transforms,

and Filters

2

Outline

• System properties• More transforms • Frequency responses and transfer functions• Motivating examples for filters• Filter terminology and basic examples• Examples of simple analog filters• Bode plots

3

Acknowledgment for Selected Slides

4

System Properties

5

System Properties

Continuous-Time System

x(t) y(t)

x[n] y[n]Discrete-Time System

• Knowing what system properties are fulfilled helps the analysis of the system and has practical implications

• Three key properties: causality, time-invariance, and linearity

6From Prof. A. S. Willsky, Signals and Systems course

Causality

7

Causality

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

8

Time-Invariance


9

Linearity


10

Additional Definitions

• LTI: Linear Time-Invariant systems• SISO: Single-Input Single-Output system• MIMO: Multi-Input Multi-Output system

In this course, we will consider only SISO filters in an operational regime respecting the LTI properties

11

More Transforms

1255

Motivation for More Transforms


Note: compare with W2, s.21

13

Laplace Transform

𝐹𝐹 𝑠𝑠 = ℒ 𝑓𝑓(𝑡𝑡) = �−∞

∞

𝑒𝑒−𝑠𝑠𝑠𝑠𝑓𝑓 𝑡𝑡 𝑑𝑑𝑡𝑡

𝑠𝑠 = 𝜎𝜎 + 𝑖𝑖𝑖𝑖

Note: the standard Laplace transform is involving an integral between 0 and ∞; the bounds - ∞ to ∞ are used for the bilateral Laplace transform which is what we will use in this course (calling by simplicity Laplace transform)

14

Fourier - Laplace

dttfesF

tftfFF

tiis

is

)(|)(

|)}({)}({)(

∫∞

∞−

−=

=

==

==

ωω

ωω L

The Fourier Transform is therefore a special case of the Laplace Transform

Fourier: frequency response (in stationary conditions, especially in signal processing)Laplace: impulse response (also in transient conditions, especially in control)

Note: non-unitary, angular frequency, see W2, s. 23

15

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 (continuous spectrum)

∑∞

−∞=

−⋅=n

nienx ωω ][)X(Note: compare with DFT notation on W2, s. 38:

X[k] = �𝑛𝑛=0

𝑁𝑁−1

x[𝑛𝑛] ⋅ 𝑒𝑒−2𝜋𝜋𝜋𝜋𝑁𝑁 𝑘𝑘𝑛𝑛

𝑘𝑘 = 0,⋯ ,𝑁𝑁 − 1

16

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 with 𝑧𝑧 = 𝑒𝑒𝒊𝒊ω (see s. 15)

𝑋𝑋(𝑧𝑧) = Z{𝑥𝑥[𝑛𝑛]} = �𝑛𝑛=−∞

∞

𝑥𝑥[𝑛𝑛]𝑧𝑧−𝑛𝑛

𝑧𝑧 = 𝐴𝐴𝑒𝑒𝒊𝒊𝜙𝜙 or 𝑧𝑧 = 𝐴𝐴(cos𝜙𝜙 + 𝒊𝒊 sin𝜙𝜙)

17

Transform Overview

Digital signalTime domain

Digital signalFrequency domain

Analog signalFrequency domain

Analog signalTime domain

Fourier/Laplace

Inverse Fourier/Laplace

DTFT/Z

Inverse DTFT/Z

DA

C

AD

C Note: Both frequency domains are continuous (spectrum)

18

Frequency Responses and Transfer Functions

19

Frequency Response

ℎ(𝑡𝑡) (ℎ ∗ 𝑓𝑓)(𝑡𝑡)𝑓𝑓(𝑡𝑡)

Time domain

𝐻𝐻(𝑖𝑖) 𝐻𝐻 𝑖𝑖 𝐹𝐹(𝑖𝑖)𝐹𝐹(𝑖𝑖)

Frequency domain

Reminder: a convolution in the time domain is a multiplication in the frequency domain

h(t) : impulse responseH(ω) : frequency response (stationary regime) of the filter/system, i.e Fourier transform of h(t)

Note: impulse response = response to an impulse𝑓𝑓 𝑡𝑡 = 𝛿𝛿(t)ℎ ∗ 𝑓𝑓 𝑡𝑡 = ℎ 𝑡𝑡

see s. 45, W2

20

Frequency Response

ℎ(𝑡𝑡)𝑒𝑒𝜋𝜋𝜔𝜔0𝑠𝑠

𝐻𝐻(𝑖𝑖) 𝐻𝐻 𝑖𝑖 2𝜋𝜋𝛿𝛿 𝑖𝑖 − 𝑖𝑖0 =𝐻𝐻(𝑖𝑖0)2𝜋𝜋𝛿𝛿 𝑖𝑖 − 𝑖𝑖0

2𝜋𝜋𝛿𝛿 𝑖𝑖 − 𝑖𝑖0

A single frequency comes out of a LTI filter multiplied with the value of the filter at that frequency.

𝐻𝐻(𝑖𝑖0)𝑒𝑒𝜋𝜋𝜔𝜔0𝑠𝑠

Note: see alsos. 12, W3

21

Frequency Response

ℎ(𝑡𝑡)𝑥𝑥 𝑡𝑡 = �𝑘𝑘=−∞

∞

𝑎𝑎𝑘𝑘𝑒𝑒𝜋𝜋𝑘𝑘𝜔𝜔0𝑠𝑠 𝑦𝑦 𝑡𝑡 = �𝑘𝑘=−∞

∞

𝐻𝐻(𝑘𝑘𝑖𝑖0)𝑎𝑎𝑘𝑘𝑒𝑒𝜋𝜋𝑘𝑘𝜔𝜔0𝑠𝑠

𝑎𝑎𝑘𝑘 → 𝐻𝐻(𝑘𝑘𝑖𝑖0)𝑎𝑎𝑘𝑘

Gain

𝐻𝐻 𝑘𝑘𝑖𝑖0 = 𝐻𝐻(𝑘𝑘𝑖𝑖0) 𝑒𝑒𝜋𝜋∡𝐻𝐻(𝑘𝑘𝜔𝜔0)

Amplitude

Phase

By linearity a sum of frequencies go out of the LTI filteronly with different amplitude and phase.

Continuous time

22

Frequency Response

ℎ[𝑛𝑛]𝑥𝑥[𝑛𝑛] = �𝑘𝑘=−∞

∞

𝑎𝑎𝑘𝑘𝑒𝑒𝜋𝜋𝑘𝑘𝜔𝜔0𝑛𝑛 𝑦𝑦[𝑛𝑛] = �𝑘𝑘=−∞

∞

𝐻𝐻(𝑘𝑘𝑖𝑖0)𝑎𝑎𝑘𝑘𝑒𝑒𝜋𝜋𝑘𝑘𝜔𝜔0𝑛𝑛

𝑎𝑎𝑘𝑘 → 𝐻𝐻(𝑘𝑘𝑖𝑖0)𝑎𝑎𝑘𝑘

Gain

𝐻𝐻 𝑘𝑘𝑖𝑖0 = 𝐻𝐻(𝑘𝑘𝑖𝑖0) 𝑒𝑒𝜋𝜋∡𝐻𝐻(𝑘𝑘𝜔𝜔0)

Amplitude

Phase

By linearity a sum of frequencies go out of the LTI filteronly with different amplitude and phase.

Discrete time

23

Time-Continuous Transfer Function

ℎ(𝑡𝑡) (ℎ ∗ 𝑓𝑓)(𝑡𝑡)𝑓𝑓(𝑡𝑡)

𝐻𝐻(𝑠𝑠) 𝐻𝐻 𝑠𝑠 𝐹𝐹(𝑠𝑠)𝐹𝐹(𝑠𝑠)

Time domain

Complex frequency domain (s-plane)

h(t) : impulse response (CT)H(s) : transfer function (stationary and transient frequency response) of the filter/system, i.e Laplace transform of h(t)


24

Time-Discrete Transfer Function

ℎ[𝑛𝑛] ℎ ∗ 𝑓𝑓 [𝑛𝑛]𝑓𝑓[𝑛𝑛]

𝐻𝐻(𝑧𝑧) 𝐻𝐻 𝑧𝑧 𝐹𝐹(𝑧𝑧)𝐹𝐹(𝑧𝑧)

Time domain

Complex frequency domain (z-plane)

h[n] : impulse response (DT)H(z) : transfer function (stationary and transient frequency response) of the filter/system, i.e. Z-transform of h[n]


25

Filters as System Examples

Analog

Analog

Circuit

Analog filter

1 1( )n ny y f x x=

Function/algorithm

Digital filter

A/D

Digital

26

Transfer Functions of FiltersAnalogCircuit

1( )1

c

in

vH sv RCs

= =+

Laplace Transf.

Numerator

Denominator

Digital

1 1( )n ny y f x x=

Function/algorithm1

0 11

1

( )1

NN

MM

b b z b zH za z b z

− −

− −

+ + +=

+ + +

Z Transf.

Numerator

Denominator

27

Motivating Examples

for Filters

2828

spectrum of original signal

spectrum of reconstructedsignal

)(tx x

( )∑+∞

−∞=

−=n

nTttp δ)(

)(txr)(ωH

spectrum of sampled signalsampling angular frequency ωs > 2 ωm

filteringfilter cut-off angular frequency ωm < ωc < (ωs –ωm)

)(txp

)()()( ωωω HXX pr =

Filters for Signal Reconstruction From W3,

s. 26

29

Reconstruction Summary –Time Domain

The reconstructed signal xr(t) is obtained through a convolutionbetween the sampled signal xp(t) with period T and one of the following three interpolation functions h(t).

Com

puta

tiona

l cos

t

1. Zero-order hold (ZOH)

2. First-order hold (FOH)

3. Whittaker-Shannon


From W3, s. 32

30

Reconstruction Summary –Frequency Domain

The spectrum of the reconstructed signal Xr(ω) is obtained through a multiplication between the spectrum of the sampled signal Xp(ω) with angular sampling frequency ωs and one of the following three low-pass filters.

From Prof. A. Oppenheim, Signals and Systems course

From W3, s. 33

31

Anti-Aliasing Filters

Reduced sampling frequency: 2 kHzAnti-alias filter: desired cut-off frequency 1 kHz

Actual high-order digital filter(see next week)

From W3, s. 42

32

Filtering Noisy Signals

low pass filter

high pass filter

Solar radiationday/night cycle

changing cloud cover

33

Terminology and Basic Filters

34

Low-Pass FilterIdealized response in the (angular) frequency domain: Notes:

• H(jω) = H(iω) = H(ω)• ωc: cut-off frequency• |H| = 1 and ∠H = 0 for

ideal filters in thepassband, no need forthe phase plot.

Realistic response in the (angular) frequency domain:

Notes: |H(jω)|: amplitude

35

High-Pass FilterIdealized response in the (angular) frequency domain:


36

Band-Pass FilterIdealized response in the (angular) frequency domain:


ωc1: lower cut-off frequencyωc2: upper cut-off frequency

Notes:a Band-Stop Filter reverses passing and stopping bands w.r.t.a band-pass filter

37

Examples of Simple Analog Filters

38

DecibelSource of sound Sound pressure Sound pressure level

pascal dB re 20 μPaJet engine at 30 m 630 Pa 150 dBRifle being fired at 1 m 200 Pa 140 dBThreshold of pain 100 Pa 130 dBHearing damage (due to short-term exposure) 20 Pa approx. 120 dB

Jet at 100 m 6 – 200 Pa 110 – 140 dBJack hammer at 1 m 2 Pa approx. 100 dBHearing damage (due to long-term exposure) 6×10−1 Pa approx. 85 dB

Major road at 10 m 2×10−1 – 6×10−1 Pa 80 – 90 dBPassenger car at 10 m 2×10−2 – 2×10−1 Pa 60 – 80 dBTV (set at home level) at 1 m 2×10−2 Pa approx. 60 dB

Normal talking at 1 m 2×10−3 – 2×10−2 Pa 40 – 60 dBVery calm room 2×10−4 – 6×10−4 Pa 20 – 30 dBLeaves rustling, calm breathing 6×10−5 Pa 10 dB

Auditory threshold at 1 kHz 2×10−5 Pa 0 dB

210

1

20 logdBVGV

=

1V 2V

𝑉𝑉2 > 𝑉𝑉1 → 𝐺𝐺𝑑𝑑𝑑𝑑 > 0 (𝑔𝑔𝑎𝑎𝑖𝑖𝑛𝑛)

𝑉𝑉2 < 𝑉𝑉1 → 𝐺𝐺𝑑𝑑𝑑𝑑 < 0 (𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑑𝑖𝑖𝑛𝑛𝑔𝑔)

39

Low Pass Filter - RC circuit

Note: -3dB = about 71% of the undamped amplitude

Break or cut-off frequency

40

High-Pass Filter – RC circuit

41

Bode Plots

42

Transfer Functions of Analog Filters

Circuit

1( )1

c

in

vH sv RCs

= =+

Laplace Transf.

Numerator

Denominator

43

From Transfer Functions to Bode Plots𝐻𝐻 𝑠𝑠 =

𝑁𝑁𝑁𝑁𝑑𝑑(𝑠𝑠)𝐷𝐷𝑒𝑒𝑛𝑛(𝑠𝑠)

= A�(𝑠𝑠 − 𝑥𝑥𝑛𝑛)𝑎𝑎𝑛𝑛(𝑠𝑠 − 𝑦𝑦𝑛𝑛)𝑏𝑏𝑛𝑛

Assume transfer function:

Frequency response: s = 𝑖𝑖𝑖𝑖

Bode magnitude:

Bode phase:

𝐻𝐻 𝑠𝑠 = 𝑖𝑖𝑖𝑖 = 𝐻𝐻 𝑖𝑖𝑖𝑖 = 𝐻𝐻 𝑖𝑖

∡𝐻𝐻 𝑠𝑠 = 𝑖𝑖𝑖𝑖 = ∡𝐻𝐻 𝑖𝑖𝑖𝑖 = ∡𝐻𝐻 𝑖𝑖

where xn, yn constants, an, bn > 0

Zero (numerator = 0): every value of s where 𝑖𝑖 = 𝑥𝑥𝑛𝑛Pole (denominator = 0): every value of s where 𝑖𝑖 = 𝑦𝑦𝑛𝑛

Bode Plot: The Example of a Low-Pass Filter

44

Bode Plot: The Example of a Low-Pass Filter

not to scale !

1000 Hz10 Hz

44

45

Bode Plots – Why handy?

𝑌𝑌 𝑖𝑖 = 𝐻𝐻 𝑖𝑖 𝑋𝑋(𝑖𝑖)𝑌𝑌 𝑖𝑖 = 𝐻𝐻 𝑖𝑖 𝑋𝑋 𝑖𝑖∡𝑌𝑌 𝑖𝑖 = ∡𝐻𝐻 𝑖𝑖 + ∡𝑋𝑋 𝑖𝑖

Frequency domain 𝐻𝐻(𝑖𝑖) 𝑌𝑌(𝑖𝑖)X(𝑖𝑖)

Amplitude

Phase

log 𝑌𝑌 𝑖𝑖 = log 𝐻𝐻 𝑖𝑖 + log 𝑋𝑋 𝑖𝑖

Note: this is valid also for cascaded blocks of LTI systems (e.g., cascaded filters)

46

Bode Plot – Why only positive and log scale for ω as well?

– If impulse response h(t) real, then 𝐻𝐻 𝑖𝑖 evenfunction of ω and ∡𝐻𝐻 𝑖𝑖 odd function of ω

– Therefore plots for negative ω can be straightforwardly obtained from those of positive ω, so disregarded

– Log scale for frequency allows for covering a wider range of possible input frequencies on the same plot

47

Bode Plot - Rules

• Zero (numerator = 0)– Amplitude: +20 dB/decade– Phase: +90º; +45º/decade, starting 1 decade

before zero• Pole (denominator = 0)

– Amplitude: -20 dB/decade– Phase: -90º; -45º/decade, starting 1 decade

before pole

48

Bode Plot (Magnitude)

Zero (numerator = 0) Amplitude: +20 dB/decadePole (denominator = 0) Amplitude: -20 dB/decade

49

Bode Plot (Phase)

Zero (numerator = 0): +90º; 45º/decade, starting 1 decade before zeroPole (denominator = 0): -90º; -45º/decade, starting 1 decade before pole

50

𝐻𝐻 𝑠𝑠 =1

1 + 𝑠𝑠𝑠𝑠𝑠𝑠

Example: Low-Pass Filter –Circuit and Analysis

)𝑑𝑑𝑣𝑣𝒐𝒐𝒐𝒐𝒐𝒐(𝑡𝑡𝑑𝑑𝑡𝑡

=1𝑠𝑠𝑠𝑠

[𝑣𝑣𝒊𝒊𝒊𝒊 𝑡𝑡 − 𝑣𝑣𝒐𝒐𝒐𝒐𝒐𝒐 𝑡𝑡 ]

𝑠𝑠𝑉𝑉𝑜𝑜𝑜𝑜𝑠𝑠(𝑠𝑠) = 1𝑅𝑅𝑅𝑅

[(𝑉𝑉𝜋𝜋𝑛𝑛(𝑠𝑠) − 𝑉𝑉𝑜𝑜𝑜𝑜𝑠𝑠(𝑠𝑠)]

𝑉𝑉𝑜𝑜𝑜𝑜𝑠𝑠(𝑠𝑠)(1 + 𝑠𝑠𝑠𝑠𝑠𝑠) = 𝑉𝑉𝜋𝜋𝑛𝑛(𝑠𝑠)

1 + 𝑠𝑠𝑠𝑠𝑠𝑠 =𝑉𝑉𝜋𝜋𝑛𝑛(𝑠𝑠)𝑉𝑉𝑜𝑜𝑜𝑜𝑠𝑠(𝑠𝑠)

=1

𝐻𝐻(𝑠𝑠)

Laplace tables & properties

ℒ ddt x t = sX(s) ℒ x t = X(s)

51

1 pole s = -1/RCi.e. 𝑖𝑖 = 𝜋𝜋

𝑅𝑅𝑅𝑅= 1

𝑅𝑅𝑅𝑅𝐻𝐻 𝑠𝑠 =1

1 + 𝑠𝑠𝑠𝑠𝑠𝑠

𝐻𝐻 𝑠𝑠 = 𝑖𝑖𝑖𝑖 = 𝐻𝐻 𝑖𝑖 =1

1 + 𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠

𝐻𝐻 𝑖𝑖 =1

1 + 𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠=

11 + 𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠

=1

1 + 𝑖𝑖2𝑠𝑠2𝑠𝑠2Bode magnitude:

Bode phase: ∡𝐻𝐻 𝑖𝑖 = tan−1𝐼𝐼𝑑𝑑[𝐻𝐻(𝑖𝑖)]𝑠𝑠𝑒𝑒[𝐻𝐻 𝑖𝑖 ]

= − tan−1 𝑖𝑖𝑠𝑠𝑠𝑠

Example: Low-Pass Filter –Circuit and Analysis

52

Example: Low-Pass Filter –Bode Plot

53

Example: High-Pass Filter –Circuit

1 zero at ω = 01 pole at ω = 1/RC

54

Example: High-Pass Filter –Bode Plot

55

Conclusion

56

Take-Home Messages• Multiple transforms exist: continuous-time Fourier, discrete-

time Fourier, Laplace, Z-transform; Laplace and Z-transform are needed to cover the large class of unstable systems

• Continuous-time Fourier transform is a special case of the Laplace transform; discrete-time Fourier transform is a special case of Z-transform

• Filters allow a number of operations (e.g., noise removal, anti-aliasing, signal reconstructions, etc.)

• Their response can be represented in time (impulse response) and frequency domain (frequency response with Fourier transform, and transfer function with Laplace transform)

• They are often easier to design and analyze in the frequency domain

• Bode plots allows for analysis of filters’ frequency response

57

Books• J. H. McClellan, R. W. Schafer, M. A. Yoder

“DSP First: A Multimedia Approach”, Prentice Hall, 1999.

• A. Oppenheim and A. S. Willsky with S. Nawab, “Signals and Systems”, Prentice Hall, 1997.

Additional Literature – Week 4

introduction to signal processing - disalw3.epfl.ch

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