Signal Fundamentals

8/12/2019 Signal Fundamentals

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Fundamentals of Signals Overview Definition

Examples

Energy and power Signal transformations

Periodic signals

Symmetry Exponential & sinusoidal signals

Basis functions

J. McNames Portland State University ECE 222 Signal Fundamentals Ver. 1.15 1


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Equation for a line

t

t0

m

x(t)

x(t) =m(t t0) You will often need to quickly write an expression for a line given

the slope and x-intercept

Will use often when discussing convolution and Fourier transforms

You should know how to apply this



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Examples of SignalsDefinition: an abstraction of any measurable quantity that is afunction of one or more independent variables such as time or space.

Examples:

A voltage in a circuit

A current in a circuit

The Dow Jones Industrial average

Electrocardiograms

A sin(t + )

Speech/music

Force exerted on a shock absorber

Concentration of Chlorine in a water supply



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Synthetic Impulse Response

0 5 10 15 20 25

1

0.5

0

0.5

1

Time (sec)



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Electrocardiogram

0 0.5 1 1.5 2 2.5

6.5

7

7.5

8

8.5

Time (sec)



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Arterial Blood Pressure

0 0.5 1 1.5 2 2.5

60

70

80

90

100

110

120

Time (sec)

ABP(mmHg)



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Speech

1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Linus: Philosophy of Wet Suckers



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Chaos

0 50 100 150 200 250 300 350 400 450

15

10

5

0

5

10

15

Time (samples)



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Discrete-time & Continuous-time We will work with both types of signals

Continuous-time signals

Will always be treated as a function oft

Parentheses will be used to denote continuous-time functions

Example: x(t)

t is a continuous independent variable (real-valued)

Discrete-time signals Will always be treated as a function ofn

Square brackets will be used to denote discrete-time functions

Example: x[n]

n is an independent integer



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Signal Energy & Power For most of this class we will use a broad definition of power and

energy that applies to any signal x(t) orx[n]

Instantaneous signal power

P(t) = |x(t)|2 P[n] = |x[n]|2

Signal energy

E(t0, t1) = t1t0|x(t)|

2

dt E(n0, n1) =

n1n=n0

|x[n]|2

Average signal power

P(t0, t1) = 1

t1 t0 t1t0|x(t)|2 dt

P(n0, n1) = 1

n1 n0+ 1

n1

n=n0|x[n]|2



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Example 1: Energy & PowerDetermine whether the energy and average power of each of thefollowing signals is finite.

x(t) =

8 |t| 0

0 otherwise

x[n] = ejn



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Example 1: Workspace (1)



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Example 1: Workspace (2)



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Signal Energy & Power Tips There are a few rules that can help you determine whether a

signal has finite energy and average power

Signals with finite energy have zero average power:

E< P= 0

Signals of finite duration and amplitude have finite energy:x(t) = 0 for|t| > c E<

Signals with finite average power have infinite energy:P>0 E=



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Signal Transformations Time shift: x(t t0) and x[n n0]

Ift0 >0 orn0 >0, signal is shifted to the right

Ift0 >0, signal appears stretched



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Example 2: Signal Transformationsx(t)

1

1 2 3 4-1

-1

-2-3-4 t

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4

Use the signal shown above to draw the following: x(t), x(t 1),x(t + 2), x( t2 ), x(2t), x(2 2t).



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Example 2: Axes for x(t + 2) & x(t

2 )

x(t)1

1 2 3 4-1

-1

-2-3-4

t

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4



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Example 2: Axes for x(2t) & x(2 2t)x(t)

1

1 2 3 4-1

-1

-2-3-4

t

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4



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Even & Odd Symmetry

xe(t) = 1

2(x(t) + x(t))

xo(t) = 1

2(x(t) x(t))

x(t) = xe(t) + xo(t)

The symmetry of a signal under time reversal will be useful laterwhen we discuss transforms

A signal is even if and only ifx(t) =x(t) A signal is odd if and only ifx(t) = x(t)

cos(k0t) is an even signal

sin(k

0t) is an odd signal

Any signal can be written as the sum of an odd signal and an evensignal



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Example 3: Even Symmetryx(t)

1

1 2 3 4-1

-1

-2-3-4 t

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4

Draw the even component of the signal shown above.



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Example 4: Odd Symmetryx(t)

1

1 2 3 4-1

-1

-2-3-4 t

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4

Draw the odd component of the signal shown above.



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Example 5: Even & Odd Symmetry

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4

t

1

1 2 3 4-1

-1

-2-3-4

Show that the sum of the even and odd components of the signal isequal to the original signal graphically.



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Periodic SignalsA signal is periodic if there is a positive value ofT orN such that

x(t) =x(t + T) x[n] =x[n + N]

The fundamental period,T0, for continuous-time signals is thesmallest positive value ofT such that x(t) =x(t + T)

The fundamental period,N0

, for discrete-time signals is thesmallest positive integer ofN such that x[n] =x[n + N]

Signals that are not periodic are said to be aperiodic



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Exponential and Sinusoidal SignalsExponential signals

x(t) =Aeat x[n] =Aean

where A and a are complex numbers.

Exponential and sinusoidal signals arise naturally in the analysis oflinear systems

Example: simple harmonic motion that you learned in physics

There are several distinct types of exponential signals

A and a real

A and a imaginary

A and a complex (most general case)



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Example 6: Aean

, A= 1 and a= 1

5

10 5 0 5 10 15 20 25 300

200

400

600

10 5 0 5 10 15 20 25 300

2

4

6

8

Time (n)



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Example 6: MATLAB Coden = -10:30; % Time indexsubplot(2,1,1);

y = exp(n/5); % Growing exponentialh = stem(n,y);set(h(1),Marker,.);

set(gca,Box,Off);subplot(2,1,2);

y = exp(-n/5); % Decaying exponentialh = stem(n,y);set(h(1),Marker,.);set(gca,Box,Off);xlabel(Time (n));



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Sinusoidal Exponential Signal Comments

x(t) =Aeat =A(ea)t =At x[n] =Aean =A(ea)n =An

When a is imaginary, then Eulers equation applies:

ejt = cos(t) + j sin(t)

ejn = cos(n) + j sin(n)

Since |ejt| = 1, this looks like a coil in a plot of the complex

plane versus time

ejt is Periodic with fundamental period T = 2

Real part is sinusoidal: Re{Aejt} =A cos(t)

Imaginary part is sinusoidal: Im{Aejt

} =A sin(t) These signals have infinite energy, but finite (constant) average

power, P



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Example 7: Aeat

, A= 1 and a=j

100

1020

30 1

0

1

1

0.5

0

0.5

1

Imaginary Part

Complex:Blue Real:Red Imaginary:Green

Time (s)

RealPart



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Example 7: MATLAB Codefs = 500; % Sample rate (Hz)t = -10:1/fs:30; % Time index (s)a = j;A = 1;y = A*exp(a*t);

h = plot3(t,imag(y),real(y),b);hold on;

h = plot3(t,ones(size(t)),real(y),r);h = plot3(t,imag(y),-ones(size(t)),g);hold off;

grid on;xlabel(Time (s));ylabel(Imaginary Part);zlabel(Real Part);title(Complex:Blue Real:Red Imaginary:Green);view(27.5,22);



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Sinusoidal Exponential Harmonics In order for ejt to be periodic with period T, we require that

ejt = ej(t+T) = ejtejT for all t

This implies ejT = 1 and therefore

T = 2k where k= 0,1,2, . . .

There is more than one frequency that satisfies the constraintx(t) =x(t + T) where T = 2k

The fundamental frequency is given by k= 1:

0= 2

T0

The other frequencies that satisfy this constraint are then integermultiples of0



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Sinusoidal Exponential Harmonics ContinuedA harmonically related set of complex exponentials is a set ofexponentials with fundamental frequencies that are all multiples of asingle positive frequency 0

k(t) = ejk0t where k= 0,1,2, . . .

For k= 0, k(t) is a constant

For all other values k(t) is periodic with fundamental frequency|k|0

This is consistent with how the term harmonic is used in music

Sinusoidal harmonics will play a very important role when wediscuss Fourier series and periodic signals



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Example 8: Continuous-Time Harmonics

1

0

1

0

1

01

1

1

0

1

2

1

0

1

3

10 5 0 5 10 15 20 25 30 35 401

01

4

Time (s)



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Example 9: Discrete-Time Harmonics

1

0

1

0

1

01

1

1

0

1

2

1

0

1

3

10 5 0 5 10 15 20 25 30 35 401

01

4

Time (n)



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Example 9: MATLAB Code

n = -10:40; % Time index

omega = 2*pi/20; % Frequency (radians/sample)

N = 4;

for cnt = 0:N,

subplot(N+1,1,cnt+1);

phi = exp(j*cnt*omega*n);

h = stem(n,real(phi));

set([h(1)],Marker,.);ylim([-1.1 1.1]);

ylabel(sprintf(\\phi_%d,cnt));

set(gca,YGrid,On)

if cnt~=N,

set(gca,XTickLabel,[]);

end;

end;

xlabel(Time (n));

figure;t = -10:0.01:40; % Time index

omega = 0.05*2*pi; % Frequency (radians/sec) - 0.05 Hz

N = 4;

for cnt = 0:N,

subplot(N+1,1,cnt+1);

phi = exp(j*cnt*omega*t);

h = plot(t,real(phi));

ylim([-1.1 1.1]);

ylabel(sprintf(\\phi_%d,cnt));grid on;

if cnt~=N,

set(gca,XTickLabel,[]);

end;

end;

xlabel(Time (s));



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Damped Complex Sinusoidal Exponentials

x(t) =Aeat x[n] =Aean

When a is complex, these becomedampedsinusoidal exponentials

Let a= +j. Then

x(t) =Aeat = (Aet) ejt x[n] =Aean = (Aen) ejn

Thus, these are equivalent to multiplying an complex sinusoid by areal exponential



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Example 10: Aean, A= 1 and a= 0.1 + j0.5

10 5 0 5 10 15 20 25 30 35 4050

0

50

Rea

lPart

10 5 0 5 10 15 20 25 30 35 402

1

0

1

2

Time (n)

RealPart



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n = -10:40; % Time index

C = 1; subplot(2,1,1);

a = 0.1 + j*0.5;

y = real(C*exp(a*n)); % Growing exponential

t = min(n):0.1:max(n);

h = plot(t, real(C*exp(real(a)*t)),t,-real(C*exp(real(a)*t)));

set(h,Color,[0.5 0.5 0.5]);

hold on;h = stem(n,y);

set(h(1),Marker,.);

hold off;

box off;

grid on;

ylim([-50 50]);

ylabel(Real Part);

subplot(2,1,2);

a = -0.1 + j*0.5;y = real(C*exp(a*n)); % Growing exponential

t = min(n):0.1:max(n);

h = plot(t, real(C*exp(real(a)*t)),t,-real(C*exp(real(a)*t)));

set(h,Color,[0.5 0.5 0.5]);

hold on;

h = stem(n,y);

set(h(1),Marker,.);

hold off;box off;

grid on;

ylim([-2 2]);

xlabel(Time (n));

ylabel(Real Part);



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Example 11: A

e

at

,A

= 1 anda

= 0.05 +

j2

10 0 10 20 30 20

22

0

2

Imaginary Part

Complex:Blue Real:Red Imaginary:Green

RealPart

10 0 10 20 30 20

22

0

2

Imaginary PartTime (s)

Re

alPart



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fs = 500; % Sample rate (Hz)

t = -10:1/fs:30; % Time index (s)

subplot(2,1,1);

a = -0.05 + j*2;

y = C*exp(a*t);

h = plot3(t,imag(y),real(y),b);

hold on;

h = plot3(t,2*ones(size(t)),real(y),r);h = plot3(t,imag(y),-2*ones(size(t)),g);

hold off;

grid on;

ylabel(Imaginary Part);

zlabel(Real Part);

title(Complex:Blue Real:Red Imaginary:Green);

axis([min(t) max(t) -2 2 -2 2]);

view(27.5,22);

subplot(2,1,2);a = +0.05 + j*2;

y = C*exp(a*t);

h = plot3(t,imag(y),real(y),b);

hold on;

h = plot3(t,2*ones(size(t)),real(y),r);

h = plot3(t,imag(y),-2*ones(size(t)),g);

hold off;

grid on;xlabel(Time (s));

ylabel(Imaginary Part);

zlabel(Real Part);

axis([min(t) max(t) -2 2 -2 2]);

view(27.5,22);



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Discrete-Time Unit Impulse

n

1

0

The discrete-time unit impulse is defined as

[n] =

0, n = 0

1, n= 0

Sometimes called the unit sample

Also called the Kroneker delta

Note that [n] has even symmetry so [n] =[n]



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Discrete-Time Basis Functions

There is a close relationship between [n] and u[n]

[n] = u[n] u[n 1]

u[n] =

nk=

[k]

u[n] =

k=0[n k]

The unit impulse can be used to sample a discrete-time signalx[n]:

x[0] =

k=

x[k

]

[k

] x

[n

] =

k=

x[k

]

[n

k]

This ability to use the unit impulse to extract a single value ofx[n]through multiplication will play an important role later in the term



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Continuous-Time Unit Step

t

u(t)

1

u(t)

0 t 0

Sometimes known as the Heaviside function

Discontinuous at t= 0

u(0) is not defined

Not of consequence because it is undefined for an infinitesimalperiod of time



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Unit Step for Switches

vs

Linear

Circuit

t=0

vsu(t)

Linear

Circuit

LinearCircuit

t=0

is

isu(t) LinearCircuit

u(t) useful for representing the opening or closing of switches We will often solve for or be given initial conditions at t= 0

We can then represent independent sources as though they wereimmediately applied at t= 0. More later.



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Continuous-Time Unit Impulse

t

ue(t)

t-e e

-e e

t

u(t)

t

1 1

e(t)

(t)

e(t) due(t)

dt

As e 0 ,

ue(t) u(t)

e(t) fort= 0 becomes very large

e(t) fort = 0 becomes zero

(t) lime0 e(t)



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Continuous-Time Unit Impulse Continued

t

1(t) (t)

0 t = 0

t= 0

ee

(t) dt= 1 for any e >0

Also known as the Dirac delta function

Is zero everywhere except zero

The impulse integral serves as a measure of the impulse amplitude

Drawn as an arrow with unit height

5(t) would be drawn as an arrow with height of 5



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Continuous-Time Unit Impulse Comments

(t) =

0 t = 0

t= 0

The impulse should be viewed as an idealization

Real systems with finite inertia do not respond instantaneously

The most important property of an impulse is its area

Most systems will respond nearly the same to sharp pulsesregardless of their shape - if

They have the same amplitude (area)

Their duration is much briefer than the systems response

The idealized unit impulse is short enough for any system



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Unit Impulse Properties

(t)x(t) = (t)x(0)

(at) = 1

|a|(t)

(t) = (t)

(t) = du(t)

dt

u(t) = t

() d



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Unit Impulse Sampling Property

+

x(t)(t) dt =

+

x(0)(t) dt

= x(0) +

(t) dt

= x(0)

Similarly, +

x(t)(t t0) dt =

+

x(t0)(t t0) dt

= x(t0) + (t t0) dt

= x(t0)



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Unit Impulse Sampling Property

x(t) =

+

x()( t) d

This integral does not appear to be useful It will turn out to be very useful

It states that x(t) can be written as a linear combination of scaledand shifted unit impulses

This will be a key concept when we discuss convolution next week



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Example 12: Continuous-Time Unit-Ramp

t

r(t)1

r(t)

0 t 0

t t 0

What is the first derivative?



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Example 13: Continuous-Time Unit-Ramp Integral

t

r(t)1

What is the integral of the unit ramp?



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Basis Function Relationships

u(t) =

t

() d

du(t)dt = (t) t

u() d = r(t)

r(t) =

t

u() d

dr(t)dt = u(t) t

r() d = 12 r(t)2

If we can write a signal x(t) in terms ofu(t) and r(t), it is easy tofind the derivative

Similarly, it is easy to integrate



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Basis Functions Translated

t

u(t-t0)

t

1 1

t

r(t-t0)

1

t0

t0

t0

(t-t0)

Can write simple expressions for the functions translated in time

Can scale the amplitude

Any piecewise linear signal can be written in terms of basisfunctions

This makes it easy to calculate derivatives and integrals

Will not discuss howthis term

Sufficient to know it can be done


Signal Fundamentals

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