Part2 Cont Signal Review

7/29/2019 Part2 Cont Signal Review

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Part 2: Review of Continuous-time Systems

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Outline

LTI systems

Impulse response

Convolution

Complex exponentials

Fourier transform

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What is an LTI system? An example

Consider the following system with microphone in a classroom

Input: x(t), speech signal

Output: y(t), speaker output

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Linear System: scaling

If I speak 2 times louder, you should hear 2 times louder voice

Originally input was x(t) and output was y(t), then if we scale theinput by a times making the input a x(t), then the output is a y(t)

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Linear System: superposition

Lecturer 1s speech is x1(t) and Lecturer 2s speech is x2(t).

Originally outputs of x1(t) and x2(t) were y1(t) and y2(t) respec-tively.

If two lecturers talk at the same time making the input x1(t)+ x2(t),

the output to audience will be y1(t) + y2(t). This is superposition.

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Time-invariant system: shifting

Again the input is x(t) and output is y(t).

For example, if the same waveform is generated 5 seconds later,x(t 5), then the output is delayed by the same amount, givingy(t 5).

For any t0, time-shift of the input by t0 causes the same amount oftime-shift at the output the output to x(t t0) is y(t t0), we callthe system as time-invariant.

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LTI system

Consider an LTI system such that for input xi(t) it outputs yi(t),i = 1, 2, . . .

If we perform scaling x

i(t)

by constant ai, shifting by time t

iand

superposition, what is the output?

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Impulse response in LTI systems

The impulse is often mathematically defined by delta function (orsometimes called sifting function)

What is the role of the delta function in LTI system? Lets see howa signal is constructed using delta functions.

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A comment on delta function

Mathematically, the delta function denoted (t) takes infinite at t =0 and is 0 elsewhere, and unit area,

(t)dt = 1

This singular property makes analysis somewhat involved. In thefollowing discussion we will use (t) instead,

(t) =

1, |t|

2

0, otherwise

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Signal construction via delta function

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Impulse response and convolution

We saw that signal x(t) is constructed by shifting, scaling andsuperposition of impulses.

So if we know the response (output) of the system by passing oneimpulse as an input, then we can generate any output signal by

proper shifting, scaling and superposition of that response. This argument holds only if the system is LTI!

Response by passing one impulse: impulse response

Shifting, scaling and superposition of impulse response adjusted toinput signal: convolution

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Convolution

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Convenient way to express sinusoidal waves.

t

Eulers formula:

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

This represents complex exponential wave with frequency .

Why use complex number? It is just a convenient way to expressand mathematically analyze. What is its meaning?

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Every function x(t) can be decomposed into even and odd parts:

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

2, xo(t) =

x(t) x(t)

2

Every even (odd) function can be represented by composition ofcosine (sine) function

Even and odd functions are orthogonal, i.e.xe(t)xo(t)dt = 0

In complex plane, real and imaginary parts are also orthogonal(inner product of vector is 0). From Eulers formula, cosine real and sine imaginary.

In Fourier analysis, even functions are related to cosine waves

and real parts of the Fourier transform.

X(j) =

x(t)ejtdt =

xe(t) cos(t)dt + j

xo(t) sin(t)dt

= XR(j) + jXI(j)

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Complex exponentials and LTI system

In LTI system, complex exponentials are eigenfunctions: if theinput is sinusoidal, the output must be sinusoidal with change only

in magnitude and phase

Consider an LTI system with impulse response h(t). When theinput is ejt the output is:

h()ej(t)d = ejt

h()ejd = H(j)ejt

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Complex exponentials and LTI system

if input is a sinusoid, output is also a sinusoid with proper shiftingand scaling

In LTI system, when the input is ejt, the output is H(j)ejt.

H(j) is the gain of a complex input sinusoidal with frequency .

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Fourier Transforms

Fourier transform (analysis)

H(j) =

h(t)ejtdt

Inverse Fourier transform (synthesis)

h(t) = 12

H(j)ejtd

Interpretation: A signal h(t) is composed of infinitely many com-plex sinusoids whose ejt component has amplitude (2)1H(j)d.

Roughly speaking this can be derived using orthogonality of com-plex exponentials with different frequency: for = ,

ejtejtdt = 0

We call h(t) and H(j) as Fourier transform pair:

h(t)F H(j)

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Phase and Magnitude

H(j) is the gain of a complex input sinusoidal with frequency .

Notation: by polar coordinate expression of complex numbers,

H(j) = |H(j)|ejH(j)

magnitude: |H(j)|

phase: H(j)

Example

h(t) = (t t0): every exponentials are shifted by same amount

t0. For example, cos(t) is shifted to cos((t t0)).

(t t0)F ejt0

In this case |H(j)| = 1, H(j) = t0.

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Convolution Theorem for LTI systems

The amplitude ofejt component ofx(t): (2)1X(j)d

The ejt component: (2)1X(j)d ejt

Going through the system, the ejt component becomes:

(2)1X(j)d ejt H(j)

(H(j) is the gain!) This is the ejt component of the output, which means

y(t) =1

2

H(j)X(j)ejtd

or

Y(j) = H(j)X(j)

This is the convolution theorem.

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sinc function

x(t) = 1, |t| < T1

0,|t|

> T1

X(j) =

T1T1

ejtdt = 2sinT1

x(t)F X(j)

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Fourier Transform Properties

Linearity: Ifx(t)F X(j) and y(t)

F Y(j) then for some

constants a and b,

ax(t) + by(t)F aX(j) + bY(j)

Even and Odd Parts: Any real signal x(t) can be decomposed into

a unique sum of even and odd functions:

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

where

xe(t) = x(t) + x(t)2

, xo(t) = x(t) x(t)2

Then

xe(t)F Re[X(j)], xo(t)

F jIm[X(j)]

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Time Scaling: For any nonzero a R

x(at)F

1

|a|X

a

W

X (W )

W

t

x (t )

t

x (t /2) 2X (2W )

This says that expanding (contracting) a signal in time contracts(expands) its spectrum.

Narrower in time broader spectrum

Broader in time narrower spectrum

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Symmetry of Inversion: Ifx(t)F X(j), then

X(t)F 2x(j)

Time Shifting: Ifx(t)F X(j) and t0 R, then

x(t t0)

F

X(j)ejt0

Frequency Shifting (modulation): If x(t)F X(j) and w0

R, then

x(t)ej0tF X(j( 0))

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Time Differentiation: Ifx(t)F X(j) then

dnx(t)

dtnF (j)nX(j)

provided that the transform exists.

Consider differentiation ofcos(t) and sin(t) with respect to t:d cos(t)

dt= sin(t) = cos(t +

2)

d sin(t)

dt

= cos(t) = sin(t +

2

)

What does this mean?

1. The magnitude of all sinusoids are multiplied by

2. the phase of all sinusoids are shifted by 2

, which is the same as

multiplying e2 = j in complex plane

This happens for each time of differentiation, thus we have to mul-tiply such gains in magnitude and phase each time : (j)n for ndifferentiation

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Time Convolution Theorem: Ifx(t), y(t)F X(j), Y(j) and

x(t)y(t) =

x(ta)y(a)da =

y(ta)x(a)da = y(t)x(t)

then

x(t) y(t)

F

X(j) Y(j).

Frequency Convolution Theorem: Ifx(t), y(t) F X(j), Y(j)then

x(t) y(t)F

1

2X(j) Y(j).

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Parsevals Theorem: Suppose x(t), y(t)F X(j), Y(j) then

x(t)y(t)dt =1

2

X(j)Y(j)d

and in particular

|x(t)|2dt =1

2

|X(j)|2d

Thus, the total energy in x(t) (think of x(t) as voltage, then itssquare gives power, and square integral over time gives energy) is

the same as the total energy in |X(j)|. So |X(j)|2 is called theenergy spectrum (the density function of energy over frequency)

ofx(t).

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Part2 Cont Signal Review

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