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7/1/14
1
MAE143A Signals & Systems http://oodgeroo.ucsd.edu/~bob/signals
Tu, Th 8:00-9:20 Pepper Canyon 106 Mo 16:00-16:50 Pepper Canyon 106
Professor Bob Bitmead [email protected] Jacobs 1609, 858-822-3477 TAs: Chun-Chia (Ben) Huang [email protected] Minh Hong Ha [email protected]
MAE143A Signals & Systems 2014 Winter 1
Classes, breaks, homeworks, midterms, final
Week Monday Tuesday Thursday 1 Jan 6 No work Jan 7 Jan 9 2 Jan 13 Jan 14 Jan 16 H1 3 Jan 20 MLKJ Jan 21 Jan 23 H2 4 Jan 27 Jan 28 M1 Jan 30 H3 5 Feb 3 Feb 4 Feb 6 H4 6 Feb 10 Feb 11 Feb 13 H5 7 Feb 17 Pres Day Feb 18 Feb 20 H6, M2 8 Feb 24 Feb 25 Feb 27 H7 9 Mar 3 Mar 4 Mar 6 H8 10 Mar 10 Mar 11 Mar 13 H9
Final Mar 20 8-11am
MAE143A Signals & Systems … ?!?!? What is this class about?
Signals – real-valued scalar functions of time x(t) Often represents a physical quantity over time
Voltage, current, pressure, speed, heart rate Could also represent economic quantities over time
Employment, value, account balance Could even represent psychological quantities
Opinions, approval ratings, confidence, satisfaction Systems – devices, processes, algorithms which operate on an input signal x(t) to produce an output signal y(t)
A system with memory is called a dynamic system
MAE143A Signals & Systems 2014 Winter 2
7/1/14
2
Signals & systems t belongs to a real interval (possibly infinite), , then
Signals x(t), y(t) are continuous-time signals System linking the two is a continuous-time system
t belongs to the natural numbers, , then Signals x(t), y(t) are discrete-time signals
Time t counts the number of sampling times, Δ System linking the two is a discrete-time system
Continuous-time dynamic systems often described by differential equations
Discrete-time dynamic systems often described by difference equations
Memory is captured by the initial conditions
MAE143A Signals & Systems 2014 Winter 3
t ∈ [a,b]
t ∈ Ν
Text Book: Luis F. Chaparro, Signals & Systems using MATLAB, Academic Press, 2011
Other related texts are fine too Some homework will refer to this exact book Web site: http://booksite.academicpress.com/chaparro/ We will stick to the book except for its close treatment of communications and control A helpful and cheap book might be the Schaum Outline Signals and Systems by Hwei Hsu, 2011 Lots of worked problems
MAE143A Signals & Systems 2014 Winter 4
7/1/14
3
Signals & Systems – (rough) planned schedule
Continuous signals and their properties 1 week Continuity, boundedness, periodicity, …
Continuous systems and their properties 3 weeks Linearity, causality, time-invariance, stability, state
Continuous signals and systems analysis 2 weeks Convolution, Fourier and Laplace transforms
Sampling & discrete time signals and systems 2 weeks Sampling, reconstruction, discrete Fourier transform (DFT)
Random signals ≤2 weeks Expectation, correlation, prediction, spectrum
MAE143A Signals & Systems 2014 Winter 5
Chapters 0,1 2, 6 3, 4, 5 7 -10
Prerequisites – what we assume you know Math 20D – Introduction to differential equations
ODEs, solutions, Laplace transforms, complex numbers Math 20E – Vector calculus
Green’s theorem, Taylor series Math 20F – Linear algebra
Matrices and vectors, bases, eigenvalues and eigenvectors MAE105 – Introduction to mathematical physics
Fourier series, integral transforms
MAE143A Signals & Systems 2014 Winter 6
7/1/14
4
Office Hours and other assistance
Bob Bitmead Wednesdays 14:00-16:00 EBU2 305
Ben Huang Mondays 17:00-18:30 EBU2 305 Minh Ha Tuesdays 15:00-17:00 EBU2 305
or by appointment
MAE143A Signals & Systems 2014 Winter 7
Homework, midterm and exam
Homework will be set weekly except Week 10 and due in class on the following Thursday The midterms will take place in class Tuesday, January 28 and Thursday February 20
sixty minutes each The final will take place Thursday, March 20, 08:00-11:00am
probably in the class room Pepper Canyon 106
MAE143A Signals & Systems 2014 Winter 8
7/1/14
5
Grading … and passing with flying colors
Final score is the maximum of the following two numbers 1.00 x Final % 0.5 x Final % + 0.3 x Midterm % + 0.2 x Homework % [Secret: the two numbers are almost always the same]
To succeed:
avail yourself of all the help including other books, past students, friends, the web
do the homework and matlab yourself seek assistance early and as necessary
MAE143A Signals & Systems 2014 Winter 9
A speech signal
MAE143A Signals & Systems 2014 Winter 10
0 1 2 3 4 5 6x 104
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Voice recording
time (s)
sam
ple
valu
e (u
nits
)
Now is the time for all good men to come to the aid of their country
7/1/14
6
Speech signal
Seven seconds of speech sampled at 22050 Hertz digitized at 16 bits A discrete-time signal representing samples from a continuous-time voltage signal which, in turn, is the output from a piezoelectric transducer of air pressure (a microphone) Because we have fairly rapid sampling we can consider (for the moment) this a continuous-time signal We will return to this later
MAE143A Signals & Systems 2014 Winter 11
0 1 2 3 4 5 6x 104
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Voice recording
time (s)
samp
le va
lue (u
nits)
Speech Signal zoomed MAE143A Signals & Systems 2014 Winter 12
0.7 0.8 0.9 1 1.1 1.2
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
’Now is’ spoken
Time (s)
spee
ch s
ampl
e (u
nits
)
N ow i s
Growing amplitudes Decaying amplitudes Low power High power Periodic high frequency low frequency Noisy/unpredictable
7/1/14
7
Speech signal
Clearly the signal is segmented (over time) into phonemes Some parts have high amplitude and therefore power
Voiced speech - most of this piece is voiced Vocal cords vibrating Strong periodic behavior Very predictable sample-to-sample
Unvoiced speech (mostly just the ‘s’ sound) Vocal cords not vibrating Mouth, lips and tongue affect the moving air Noisy looking Not predictable sample-to-sample
We see different rates of attack and decay Can you identify the Australian accent?
MAE143A Signals & Systems 2014 Winter 13
Familiar signals – the constant signal
The constant signal constant over all time
The Laplace transform of this constant signal is Note that the Laplace transform ignores the t<0 part
MAE143A Signals & Systems 2014 Winter 14
x(t) = c, t ∈ (−∞,∞)
Lx
(s) =
Z 1
0�x(t)e�st
dt =0.7
�s
e
�st
����1
0�=
0.7
s
7/1/14
8
Familiar signals – the step function
The unit step function or Heaviside function The Laplace transform of the unit step This is the same as for a constant function of value 1 because they are identical for t ≥0 This function is discontinuous at t=0
MAE143A Signals & Systems 2014 Winter 15
1(t) =
(0, t < 0
1, t � 0
L1(s) =
Z 1
0�1(t)e�st dt =
1
s
Continuous approximation to a unit step
Here are some approximations to 1(t) which are continuous Here erf(x) is the error function The red curve is k=10 The black curve is k=20
MAE143A Signals & Systems 2014 Winter 16
1̂k(t) =1
2+
1
2erf(kt)
erf(x) =
2p⇡
Zx
�1exp(�z
2) dz
7/1/14
9
Familiar signals – the ramp function
Ramp function
This function is unbounded but it is continuous Laplace transform Notice that the ramp function is the integral of the step
MAE143A Signals & Systems 2014 Winter 17
r(t) =
(0, t < 0
t, t � 0
Lr(s) =
Z 1
0�r(t)e�st dt =
Z 1
0�te�st dt =
1
s2
r(t) =
Z t
�11(z) dz
Familiar signals – the impulse function The impulse function or Dirac delta function The impulse function is neither continuous nor bounded
Laplace transform
The step is the integral of the impulse
MAE143A Signals & Systems 2014 Winter 18
�(t) = 0, for t 6= 0
R ✏�✏ �(t) dt = 1, for ✏ > 0
L�(s) =
Z 1
0��(t)e�st dt =
Z 0+
0��(t) dt = 1
1(t) =
Z t
�1�(z) dz
7/1/14
10
Bounded continuous approximation of the impulse Continuous and bounded approximations of δ(t) red is black is
The impulse function has a sampling property
for any function f(t) continuous at t=0
MAE143A Signals & Systems 2014 Winter 19
�̂k(t) = ksin kt
t
17.5⇥ �̂40(t) = 700sin 40t
t
25⇥ �̂20(t) = 500sin 20t
t
Z b
af(z)�(z) dz = f(0) if 0 2 (a, b)
Familiar signals – real exponentials
Red Blue Black
Laplace Red Blue Black
Laplace
MAE143A Signals & Systems 2014 Winter 20
e0.5t
et
e2t
e�2te�te�0.5t
All of these signals are unbounded
1
s� 0.5,
1
s� 1,
1
s� 2
1
s+ 0.5,
1
s+ 1,
1
s+ 2
7/1/14
11
Familiar signals – one-sided real exponentials Red Blue Black
Laplace
poles -0.5, -1, -2 Red Blue Black
Laplace
poles (0,0.5), (0,1), (0,2)
MAE143A Signals & Systems 2014 Winter 21
e�0.5t1(t)e�t1(t)
e�2t1(t)
1
s� 0.5� 1
s,
1
s� 1� 1
s,
1
s� 2� 1
s
⇥e0.5t � 1
⇤1(t)⇥
et � 1⇤1(t)⇥
e2t � 1⇤1(t)
1
s+ 0.5
1
s+ 1
1
s+ 2
Familiar signals - sinusoids
Red Blue Black Red Black
MAE143A Signals & Systems 2014 Winter 22
sin(5t)
sin(3t)
sin(7t)
cos(5t)
sin(5t)
7/1/14
12
One-sided sinusoids Sinusoids Laplace transforms Poles ±j10, ±j15, ±j20 Sinusoid and cosinusoid Laplace transforms Poles ±j10
MAE143A Signals & Systems 2014 Winter 23
sin(10t)1(t) sin(15t)1(t) sin(20t)1(t)
10
s2 + 100
15
s2 + 225
20
s2 + 400
sin(10t)1(t) cos(10t)1(t)
10
s2 + 100
s
s2 + 100
Complex exponentials
Blue Red Laplace transforms Poles -2±j20, -2 The (upper) red curve is called the envelope of the blue curve
MAE143A Signals & Systems 2014 Winter 24
e�2t sin(20t)1(t)
20
(s+ 2)2 + 202=
20
s2 + 4s+ 4041
s+ 2
±e�2t1(t)
7/1/14
13
More complex exponentials
Blue Red
Laplace transform
Poles 2±j20, 2 in the right half of the complex plane
that is, the real part is positive These signals are unbounded
MAE143A Signals & Systems 2014 Winter 25
e2t sin(20t)1(t)e2t1(t)
20
(s� 2)2 + 202=
20
s2 � 4s+ 404
1
s� 2
Periodic signals
Periodic signals repeat The minimal cycle time T is called the period Here it is one second For a periodic signal we only need to specify it over one period and we know it everywhere Sinusoids, cosinusoids and constants are periodic One-sided variants are not, k above can be negative
MAE143A Signals & Systems 2014 Winter 26
x(t+ kT ) = x(t) for k 2 Z
7/1/14
14
Even and odd signals
Even signals such as cos(t)
Odd signals
such as sin(t)
MAE143A Signals & Systems 2014 Winter 27
x(�t) = x(t)
x(�t) = �x(t)
x(t) = x
even
(t) + x
odd
(t)
x
even
(t) =1
2[x(t) + x(�t)]
x
odd
(t)1
2[x(t)� x(�t)]
Real world industrial signals
Macknade bulk sugar dryer, Queensland Australia 11m-long rotating drum evaporative cooling and drying
Hot wet sugar in the top (left), cool dry air in the bottom (right) Hot moist air out the top, cool dry sugar out the bottom
MAE143A Signals & Systems 2013 Winter 28
Sugar InSugarOutAir OutAir In
7/1/14
15
Macknade rotary bulk sugar dryer - experiments
MAE143A Signals & Systems 2014 Winter 29
Input sugar temperature signal
MAE143A Signals & Systems 2014 Winter 30
7/1/14
16
Input air temperature signal
MAE143A Signals & Systems 2014 Winter 31
Input air humidity signal
MAE143A Signals & Systems 2014 Winter 32
7/1/14
17
Output sugar temperature signal
MAE143A Signals & Systems 2014 Winter 33
A quantized signal
Macknade sugar dryer is a 3-input 1-output system
MAE143A Signals & Systems 2014 Winter 34
7/1/14
18
Mathematical model of the sugar dryer system
This is a set of nonlinear difference equations in (state) variables Ei, Ms
i, Mai, Mm
i, Mvi,Ta
i, Tsi
The blue quantities are parameters of the model
MAE143A Signals & Systems 2014 Winter 35
s a v
M i s ( k τ ) = M i
s ([ k - 1 ] τ ) - Ei ( k τ ), M i a ( k τ ) = M i
a ([ k - 1 ] τ ) + E i ( k τ )
T i a ( k τ ) = T i
a ([ k - 1 ] τ ) + hA τ + C pv E i ( k τ ) [ ] T i
s ([ k - 1 ] τ ) - T i a ([ k - 1 ] τ ) [ ]
C pa M i a ( k τ ) + C pv M i
v ( k τ )
T i s ( k τ ) = T i
s ([ k - 1 ] τ ) - L H 2 O E i ( k τ ) + hA τ T i
s ([ k - 1 ] τ ) - T i a ([ k - 1 ] τ ) [ ]
C ps M i s ( k τ ) + C pw M i
w ( k τ )
M i m ( k τ ) = [ 1 - α ] M i
m ( k τ ) + α M i - 1 m ( k τ ), M i
v ( k τ ) = M i + 1 v ( k τ )
E i ( k τ ) = mA τ ( P sugar ([ k - 1 ] τ , T i ) - P air ([ k - 1 ] τ , M i , M i ))
Properties of signals
Domain – region of times under consideration Continuous-time, discrete-time, finite time interval …
Support – region of time over which they are nonzero One-sided functions, impulses, limited extent
Amplitude – maximal magnitude Boundedness, norm, energy
Smoothness – degree of continuity, differentiability, etc. Everywhere, piecewise, …
Periodic, deterministic, random, etc. Generally connected with the ability to predict the signal
MAE143A Signals & Systems 2014 Winter 36
7/1/14
19
Signal norms – a measure of signal size
General Lp signal norms L2 Euclidean signal norm
often related to signal energy voltage, current, velocity signals
L1 norm L∞ norm or sup norm
MAE143A Signals & Systems 2014 Winter 37
kxkp4=
Z 1
�1|x(t)|p dt
� 1p
kxk24=
Z 1
�1|x(t)|2 dt
� 12
kxk14=
Z 1
�1|x(t)| dt
kxk14= sup
t2(�1,1)|x(t)|
Signal transforms – Laplace and Fourier
Expression of the signal in a different domain Laplace transform for signals defined on domain [0-,∞] Fourier transform for signals defined on the domain [-∞, ∞] Since the transforms are invertible no information is lost in
using them instead of the original time-domain description
MAE143A Signals & Systems 2014 Winter 38
X(!)
4=
Z 1
�1x(t)e
�j!tdt, x(t) =
1
2⇡
Z 1
�1X(!)e
j!td!, for all t
X(s)
4=
Z 1
0�x(t)e
�stdt
x(t) =
1
2⇡j
Z c+j1
c�j1X(s)e
stds, for t � 0