introduction to signal processing - disalw3.epfl.ch
TRANSCRIPT
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
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
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
1255
Motivation for More Transforms
From Prof. A. S. Willsky, Signals and Systems course
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)
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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
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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 (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
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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 with 𝑧𝑧 = 𝑒𝑒𝒊𝒊ω (see s. 15)
𝑋𝑋(𝑧𝑧) = Z{𝑥𝑥[𝑛𝑛]} = �𝑛𝑛=−∞
∞
𝑥𝑥[𝑛𝑛]𝑧𝑧−𝑛𝑛
𝑧𝑧 = 𝐴𝐴𝑒𝑒𝒊𝒊𝜙𝜙 or 𝑧𝑧 = 𝐴𝐴(cos𝜙𝜙 + 𝒊𝒊 sin𝜙𝜙)
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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)
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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
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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
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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
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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
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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)
Reminder: a convolution in the time domain is a multiplication in the frequency domain
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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]
Reminder: a convolution in the time domain is a multiplication in the frequency domain
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Filters as System Examples
Analog
Analog
Circuit
Analog filter
1 1( )n ny y f x x=
Function/algorithm
Digital filter
A/D
Digital
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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
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 Prof. A. S. Willsky, Signals and Systems course
From W3, s. 32
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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
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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
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:
Realistic response in the (angular) frequency domain:
36
Band-Pass FilterIdealized response in the (angular) frequency domain:
Realistic 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
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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
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
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
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