MAE143A Signals & Systems … ?!?!?numbat.ucsd.edu/~bob/signals2014Winter/Signals_&_Systems_Winter... · 7/1/14 3 Signals & Systems – (rough) planned schedule Continuous signals

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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


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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


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


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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


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


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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


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


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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


A speech signal


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

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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


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

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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?


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


x(t) = c, t ∈ (−∞,∞)

Lx

(s) =

Z 1

0�x(t)e�st

dt =0.7

�s

e

�st

��1

0�=

0.7

s

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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


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


1̂k(t) =1

2+

1

2erf(kt)

erf(x) =

2p⇡

Zx

�1exp(�z

2) dz

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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


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


�(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

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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


�̂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


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

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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)


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


sin(5t)

sin(3t)

sin(7t)

cos(5t)

sin(5t)

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One-sided sinusoids Sinusoids Laplace transforms Poles ±j10, ±j15, ±j20 Sinusoid and cosinusoid Laplace transforms Poles ±j10


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


e�2t sin(20t)1(t)

20

(s+ 2)2 + 202=

20

s2 + 4s+ 4041

s+ 2

±e�2t1(t)

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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


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


x(t+ kT ) = x(t) for k 2 Z

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Even and odd signals

Even signals such as cos(t)

Odd signals

such as sin(t)


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


Sugar InSugarOutAir OutAir In

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Macknade rotary bulk sugar dryer - experiments


Input sugar temperature signal


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Input air temperature signal


Input air humidity signal


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Output sugar temperature signal


A quantized signal

Macknade sugar dryer is a 3-input 1-output system


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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


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


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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


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


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

Documents

MAE143A Signals & Systems … ?!?!?numbat.ucsd.edu/~bob/signals2014Winter/Signals_&_Systems_Winter... · 7/1/14 3 Signals & Systems – (rough) planned schedule Continuous signals