Lecture 8 Properties of the Fourier Transform

Lecture 8 ELE 301: Signals and Systems

Prof. Paul Cuff

Princeton University

Fall 2011-12

Cuff (Lecture 7)

ELE 301: Signals and Systems

Properties of the Fourier Transform

Properties of the Fourier Transform Linearity Time-shift Time Scaling Conjugation Duality Parseval

Convolution and Modulation

Periodic Signals

Constant-Coefficient Differential Equations

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 1 / 37 Fall 2011-12 2 / 37

Linearity

Linear combination of two signals x1(t) and x2(t) is a signal of the form ax1(t) + bx2(t). Linearity Theorem: The Fourier transform is linear; that is, given two signals x1(t) and x2(t) and two complex numbers a and b, then

ax1(t) + bx2(t) aX1(j) + bX2(j).

This follows from linearity of integrals:

(ax1(t) + bx2(t))e-j2ft dt

-

=a

x1(t)e-j2ft dt + b

-

= aX1(f ) + bX2(f )

x2(t)e-j2ft dt

-

Cuff (Lecture 7)

Finite Sums

ELE 301: Signals and Systems

Fall 2011-12 3 / 37

This easily extends to finite combinations. Given signals xk (t) with Fourier transforms Xk (f ) and complex constants ak , k = 1, 2, . . . K , then

K

K

ak xk (t) ak Xk (f ).

k =1

k =1

If you consider a system which has a signal x(t) as its input and the Fourier transform X (f ) as its output, the system is linear!

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 4 / 37

Linearity Example

Find the Fourier transform of the signal

x(t) =

1 2

1 2

|t |

<

1

1

|t |

1 2

This signal can be recognized as

1

t1

x(t) = rect + rect (t)

2

22

and hence from linearity we have

1

1

1

X (f ) = 2 sinc(2f ) + sinc(f ) = sinc(2f ) + sinc(f )

2

2

2

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 5 / 37

1.2

1

1rect(t/2) + 1rect(t)

0.8

2

2

0.6

0.4

0.2

0

!0.2

-2 -1 !2.5

!2

!1.5

!1

!0.5

00

0.5

11

1.5

22

2.5

2 1.5

1 0.5

0 !0.5

!10 -!48 !6

-!42 !2

sinc(/) +

1 2

sinc(/(2))

00

2

24 6

48 10

Cuff (Lecture 7)

ELE 301: Signals and Systems

L

Fall 2011-12 6 / 37

Scaling Theorem

Stretch (Scaling) Theorem: Given a transform pair x(t) X (f ), and a real-valued nonzero constant a,

1f x(at) X

|a| a

Proof: Here consider only a > 0. (negative a left as an exercise) Change variables = at

x (at)e-j2ft dt =

x ( )e-j2f /a d

=

1 X

f

.

-

-

aaa

If a = -1 "time reversal theorem:" X (-t) X (-f )

Cuff (Lecture 7)

Scaling Examples

ELE 301: Signals and Systems

Fall 2011-12 7 / 37

We have already seen that rect(t/T ) T sinc(Tf ) by brute force integration. The scaling theorem provides a shortcut proof given the simpler result rect(t) sinc(f ).

This is a good point to illustrate a property of transform pairs. Consider this Fourier transform pair for a small T and large T , say T = 1 and T = 5. The resulting transform pairs are shown below to a common horizontal scale:

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 8 / 37

Compress in time - Expand in frequency

1.2 1

0.8 0.6 0.4 0.2

0 !0.2

-!1200

-!105

rect(t)

00

150 1020

t

6

sinc(/2)

4 2 0 !2

-!1100 -!55 00 55 1100

1.2 1

0.8 0.6 0.4 0.2

0 !0.2

-!1200

-!105

rect(t/5)

00

150 1020

t

5 4 3 2 1 0 !1 !2

-!1100 -!55

5sinc(5/2) 00 55 1100

NarrowCueffr (pLeuctlusree7)means higherELbEa3n01d: wSigindalts han.d Systems

Scaling Example 2

Fall 2011-12 9 / 37

As another example, find the transform of the time-reversed exponential

x(t) = eat u(-t).

This is the exponential signal y (t) = e-atu(t) with time scaled by -1, so the Fourier transform is

1

X (f ) = Y (-f ) =

.

a - j2f

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 10 / 37

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download