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# Lecture 1: Introduction to Signals

Summary: To describe signals, both man-made and naturally occurring, in forms of mathematical expressions in time and frequency domains.

Lecture #1:

INTRODUCTION TO SIGNALS

Motivation: To describe signals, both man-made and naturally occurring, in forms of mathematical expressions in time and frequency domains.

Outline:

• Classification of signals
• Building-block signals — complex exponentials, impulses.
• Conclusions

Signals and systems

This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.

• Signals are variables that carry information
• Systems process input signals to produce output signals.

Today — SIGNALS; Next time — SYSTEMS.

Demonstration of different types of signals

I. CLASSIFICATION OF SIGNALS

Identity of the independent variable

Time is often the independent variable for signals. For example, the electrical activity of the heart recorded with electrodes on the surface of the chest — the electrocardiogram (ECG or EKG).

Generic time

The term time is often used generically to represent the independent variable of a signal. The independent variable may be a spatial variable as in an image. Here color information is specified as a function of position.

Dimensionality of the independent variable

The independent variable can be 1-D (time t in the EKG signal) or 2-D (space x, y in the image), 3-D, or N-D.

In this course, we shall consider largely 1-D signals, but signals in many applications (e.g., radio astronomy, medical imaging, seismometry) have multiple dimensions.

1. Continuous time (CT) and discrete time (DT) signals

CT signals take on real or complex values as a function of an independent variable that ranges over the real numbers and are denoted as x(t). DT signals take on real or complex values as a function of an independent variable that ranges over the integers and are denoted as x[n]. Note the subtle use of parentheses and square brackets to distinguish between CT and DT signals.

For example, consider the image shown on the left and its DT representation shown on the right

The image on the left consists of 302 × 435 picture elements (pixels) each of which is represented by a triplet of numbers {R,G,B} that encode the color. Thus, the signal is represented by c[n,m] where m and n are the independent variables that specify pixel location and c is a color vector specified by a triplet of hues {R,G,B} (red, green, and blue).

2/ Real and complex signals

Signals can be real, imaginary, or complex. An important class of signals are the complex exponentials:

• the CT signal x(t) = estest size 12{e rSup { size 8{ ital "st"} } } {} where is s is a complex number,
• the DT signal x[n] = znzn size 12{z rSup { size 8{n} } } {} where z is a complex number.

Q. Why do we deal with complex signals?

A. They are often analytically simpler to deal with than real signals.

For both exponential CT (x(t) = estest size 12{e rSup { size 8{ ital "st"} } } {}) and DT (x[n] = znzn size 12{z rSup { size 8{n} } } {}) signals, x is a complex quantity. To plot x, we can choose to plot either its magnitude and angle or its real and imaginary parts - whichever is more convenient for the analysis.

For example, suppose s = jπ/8

and z = e/8e/8 size 12{e rSup { size 8{jπ/8} } } {}, then the real parts R

are:

R{x(t)} = R{ e/8e/8 size 12{e rSup { size 8{jπ/8} } } {}} = cos(πt/8)

R{x[n]} = R{ e/8e/8 size 12{e rSup { size 8{jπ/8} } } {}} = cos[πn/8]

3/ Periodic and aperiodic signals

Periodic signals have the property that x(t + T) = x(t) for all t. The smallest value of T that satisfies the definition is called the period. Shown below are an aperiodic signal (left) and a periodic signal (right).

4/ Causal and anti-causal signals

A causal signal is zero for t < 0 and an anti-causal signal is zero for t > 0

5/ Right- and left-sided signals

A right-sided signal is zero for t < T and a left-sided signal is zero for t > T where T can be positive or negative.

6/ Bounded and unbounded signals

7/ Even and odd signals

Even signals xe(t) and odd signals xo(t) are defined as

{ x e ( t ) = x e ( t ) x o ( t ) = x o ( t ) { x e ( t ) = x e ( t ) x o ( t ) = x o ( t ) size 12{ left lbrace matrix { x rSub { size 8{e} } $$t$$ =x rSub { size 8{e} } $$- t$$  {} ## x rSub { size 8{o} } $$t$$ = - x rSub { size 8{o} } $$- t$$ } right none } {}

Any signal is a sum of unique odd and even signals. Using

{ x ( t ) = x e ( t ) + x o ( t ) x ( t ) = x e ( t ) x o ( t ) { x ( t ) = x e ( t ) + x o ( t ) x ( t ) = x e ( t ) x o ( t ) size 12{ left lbrace matrix { x $$t$$ =x rSub { size 8{e} } $$t$$ +x rSub { size 8{o} } $$t$$ {} ## x $$- t$$ =x rSub { size 8{e} } $$t$$ - x rSub { size 8{o} } $$t$$ } right none } {}

Yields

{ x e ( t ) = 1 2 ( x ( t ) + x ( t ) ) x o ( t ) = 1 2 ( x ( t ) x ( t ) ) { x e ( t ) = 1 2 ( x ( t ) + x ( t ) ) x o ( t ) = 1 2 ( x ( t ) x ( t ) ) size 12{ left lbrace matrix { x rSub { size 8{e} } $$t$$ = { {1} over {2} } $$x \( t$$ +x $$- t$$ \) {} ## x rSub { size 8{o} } $$t$$ = { {1} over {2} } $$x \( t$$ - x $$- t$$ \) } right none } {}

II. BUILDING-BLOCK SIGNALS

We will represent signals as sums of building-block signals. Important families of building-block signals are the eternal, complex exponentials and the unit impulse functions.

1/ Eternal, complex exponentials

These signals have the form x(t) = X estest size 12{e rSup { size 8{ ital "st"} } } {} for all t and x[n] = X znzn size 12{z rSup { size 8{n} } } {} for all n, where X, s, and z are complex numbers. We illustrate the richness of this class of functions for CT signals; DT signals are similarly rich. In general s is complex and can be written as s = σ +jω, where σ and ω are the real and imaginary parts of s.

a/ Eternal, complex exponentials — real s

If s = σ is real and X is real then

x ( t ) = Xe σt x ( t ) = Xe σt size 12{x $$t$$ = ital "Xe" rSup { size 8{σt} } } {}

and we get the family of real exponential functions.

b/ Eternal, complex exponentials — imaginary s

If s = jω is imaginary and X is real then

x ( t ) = Xe jϖt = X ( cos ( ϖt ) + j sin ( ϖt ) ) x ( t ) = Xe jϖt = X ( cos ( ϖt ) + j sin ( ϖt ) ) size 12{x $$t$$ = ital "Xe" rSup { size 8{jϖt} } =X $$"cos" \( ϖt$$ +j"sin" $$ϖt$$ \) } {}

and we get the family of sinusoidal functions.

c/ Eternal, complex exponentials — complex s

If s = σ +jω is complex and X is real then

x ( t ) = Xe ( σ + ) t = Xe σt ( cos ( ϖt ) + j sin ( ϖt ) ) x ( t ) = Xe ( σ + ) t = Xe σt ( cos ( ϖt ) + j sin ( ϖt ) ) size 12{x $$t$$ = ital "Xe" rSup { size 8{ $$σ+jϖ$$ t} } = ital "Xe" rSup { size 8{σt} } $$"cos" \( ϖt$$ +j"sin" $$ϖt$$ \) } {} {}

and we get the family of damped sinusoidal functions.

For x(t) = X estest size 12{e rSup { size 8{ ital "st"} } } {}, R{x(t)}= X eσteσt size 12{e rSup { size 8{σt} } } {}cosωt is plotted for different values of s superimposed on the complex s-plane.

For x(t) = X estest size 12{e rSup { size 8{ ital "st"} } } {}, I {x(t)} = X estest size 12{e rSup { size 8{ ital "st"} } } {} sinωt is plotted for different values of s superimposed on the complex s-plane.

d/ Eternal complex exponentials — why are they important?

• Almost any signal of practical interest can be represented as a superposition (sum) of eternal complex exponentials.
• The output of a linear, time-invariant (LTI) system (to be defined next time) is simple to compute if the input is a sum of eternal complex exponentials.
• Eternal complex exponentials are the eigenfunctions or characteristic (unforced, homogeneous) responses of LTI systems.

2/ Unit impulse — definition

The unit impulse δ(t), called the Dirac delta function, is not a function in the ordinary sense. It is defined by the integral relation

f ( t ) δ ( t ) dt = f ( 0 ) f ( t ) δ ( t ) dt = f ( 0 ) size 12{ Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {f $$t$$ δ $$t$$ ital "dt"=f $$0$$ } } {}

and is called a generalized function.

The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t). To see that the area of the unit impulse is 1, choose f(t) = 1 in the definition. We represent the unit impulse schematically as shown below; the number next to the impulse is its area.

a/ Unit impulse — narrow pulse approximation

To obtain an intuitive feeling for the unit impulse, it is often helpful to imagine a set of rectangular pulses where each pulse has width ε and height 1/ ε so that its area is 1.

The unit impulse is the quintessential tall and narrow pulse!

b/ Unit impulse — intuiting the definition

To obtain some intuition about the meaning of the integral definition of the impulse, we will use a tall rectangular pulse of unit area as an approximation to the unit impulse.

As the rectangular pulse gets taller and narrower,

lim ε 0 f ( t ) p ε ( t ) dt f ( 0 ) ε . ε = f ( 0 ) lim ε 0 f ( t ) p ε ( t ) dt f ( 0 ) ε . ε = f ( 0 ) size 12{ {"lim"} cSub { size 8{ε rightarrow 0} } Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {f $$t$$ p rSub { size 8{ε} } $$t$$ ital "dt" rightarrow  { {f $$0$$ } over {ε} } "." ε=f $$0$$ } } {}

c/ Unit impulse — the shape does not matter

There is nothing special about the rectangular pulse approximation to the unit impulse. A triangular pulse approximation is just as good. As far as our definition is concerned both the rectangular and triangular pulse are equally good approximations. Both act as impulses.

d/ Unit impulse — the values do not matter

The values of the approximating functions do not matter either. The function on the left has unit area and takes on the arbitrary value A for t = 0. The function on the right, which we shall encounter frequently in later lectures, has the property that it has non-zero values at most of its values, all but a countably infinite number of points, but still acts as a unit impulse.

What all these approximations have in common is that as ε gets small the area of each function occupies an increasingly narrow time interval centered on t = 0.

Two-minute miniquiz problem

Problem 1-1

Interpret and sketch the generalized function x(t) where

x ( t ) = e t / 4 δ ( t + 4 ) x ( t ) = e t / 4 δ ( t + 4 ) size 12{x $$t$$ =e rSup { size 8{t/4} } δ $$t+4$$ } {}

Solution

To determine the meaning of x(t) we place it in an integral

e t / 4 δ ( t + 4 ) dt = e ( τ 4 ) / 4 δ ( τ ) e t / 4 δ ( t + 4 ) dt = e ( τ 4 ) / 4 δ ( τ ) size 12{ Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {e rSup { size 8{t/4} } δ $$t+4$$ ital "dt"={}} Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {e rSup { size 8{ $$τ - 4$$ /4} } δ $$τ$$ dτ} } {}

From the definition of the unit impulse,

f ( t ) δ ( t ) dt = f ( 0 ) f ( t ) δ ( t ) dt = f ( 0 ) size 12{ Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {f $$t$$ δ $$t$$ ital "dt"=f $$0$$ } } {}

the integral equals e1e1 size 12{e rSup { size 8{ - 1} } } {}. Therefore,

x ( t ) = e 1 δ ( t + 4 ) x ( t ) = e 1 δ ( t + 4 ) size 12{x $$t$$ =e rSup { size 8{ - 1} } δ $$t+4$$ } {}

The result can also be seen graphically. The left panel shows both et/4et/4 size 12{e rSup { size 8{t/4} } } {} and δ(t+4), and the right panel shows their product.

e/ Unit impulse — what do we need it for?

The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations.

• Impulse of current in time delivers a unit charge instantaneously to a network.
• Impulse of force in time delivers an instantaneous momentum to a mechanical system.
• Impulses in space are also useful.
• Impulse of mass density in space represents a point mass.
• Impulse of charge density in space represents a point charge.
• Impulse of light intensity in space represents a point of light.

We can imagine impulses in space and time.

• Impulse of light intensity in space and time represents a brief flash of light at a point in space.

f/ Unit step

Integration of the unit impulse yields the unit step function

u ( t ) = t δ ( τ ) u ( t ) = t δ ( τ ) size 12{u $$t$$ = Int rSub { size 8{ - infinity } } rSup { size 8{t} } {δ $$τ$$ dτ} } {}

which is defined as

u ( t ) = { 0 ( t < 0 ) 1 ( t 0 ) u ( t ) = { 0 ( t < 0 ) 1 ( t 0 ) size 12{u $$t$$ = left lbrace matrix { 0 $$t<0$$ {} ## 1 $$t >= 0$$ } right none } {}

g/ Unit impulse as the derivative of the unit step

As an example of the method for dealing with generalized functions consider the generalized function

x ( t ) = d dt u ( t ) x ( t ) = d dt u ( t ) size 12{x $$t$$ = { {d} over { ital "dt"} } u $$t$$ } {}

Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function

y ( t ) = f ( t ) d dt u ( t ) dt y ( t ) = f ( t ) d dt u ( t ) dt size 12{y $$t$$ = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {f $$t$$ { {d} over { ital "dt"} } u $$t$$ ital "dt"} } {}

and apply the usual integration-by-parts theorem

y ( t ) = f ( t ) u ( t ) u ( t ) d dt f ( t ) dt y ( t ) = f ( t ) u ( t ) u ( t ) d dt f ( t ) dt size 12{y $$t$$ =f $$t$$ u $$t$$ \lline matrix { infinity {} ## - infinity }  -  Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {u $$t$$ { {d} over { ital "dt"} } f $$t$$ ital "dt"} } {}

to obtain

y ( t ) = f ( ) 0 d dt f ( t ) dt = f ( 0 ) y ( t ) = f ( ) 0 d dt f ( t ) dt = f ( 0 ) size 12{y $$t$$ =f $$infinity$$  -  Int rSub { size 8{0} } rSup { size 8{ infinity } } { { {d} over { ital "dt"} } f $$t$$ ital "dt"} =`f $$0$$ } {}

The result is that

f ( t ) d dt u ( t ) dt = 0 f ( t ) d dt u ( t ) dt = 0 size 12{ Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {f $$t$$ { {d} over { ital "dt"} } u $$t$$ ital "dt"=0} } {}

which, from the definition of the unit impulse, implies that

δ ( t ) = d dt u ( t ) δ ( t ) = d dt u ( t ) size 12{δ $$t$$ = { {d} over { ital "dt"} } u $$t$$ } {}

That is, the unit impulse is the derivative of the unit step in a generalized function sense.

h/ Successive integration of the unit impulse

Successive integration of the unit impulse yields a family of functions.

Later we will talk about the successive derivatives of δ(t), but these are too horrible to contemplate in the first lecture.

3/ Building-block signals can be combined to make a rich population of signals

Eternal complex exponentials and unit steps can be combined to produce causal and anti-causal decaying exponentials

Unit steps and unit ramps can be combined to produce pulse signals.

III. CONCLUSIONS

• We are awash in a sea of signals.
• Signal categories — identity of independent variable, dimensionality, CT & DT, real & complex, periodic & aperiodic, causal & anti-causal, bounded & unbounded, even & odd, etc.
• Building block signals — eternal complex exponentials and impulse functions — are a rich class of signals that can be superimposed to represent virtually any signal of physical interest.

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