Lecture 8 Properties of the Fourier Transform

[Pages:19]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

Scaling Example 3

As a final example which brings two Fourier theorems into use, find the

transform of

x (t) = e-a|t|.

This signal can be written as e-atu(t) + eatu(-t). Linearity and time-reversal yield

1

1

X (f ) =

+

a + j2f a - j2f

2a = a2 - (j2f )2

2a = a2 + (2f )2

Much easier than direct integration!

Cuff (Lecture 7)

ELE 301: Signals and Systems

Complex Conjugation Theorem

Fall 2011-12 11 / 37

Complex Conjugation Theorem: If x(t) X (f ), then x(t) X (-f )

Proof: The Fourier transform of x(t) is

x (t)e-j2ft dt =

-

=

x (t)ej2ft dt

-

x (t)e-(-j2f )t dt

-

= X (-f )

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 12 / 37

Duality Theorem

We discussed duality in a previous lecture. Duality Theorem: If x(t) X (f ), then X (t) x(-f ). This result effectively gives us two transform pairs for every transform we find.

Exercise What signal x(t) has a Fourier transform e-|f |?

Cuff (Lecture 7)

Shift Theorem

ELE 301: Signals and Systems

Fall 2011-12 13 / 37

The Shift Theorem: Proof:

x (t - ) e-j2f X (f )

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 14 / 37

Example: square pulse

Consider a causal square pulse p(t) = 1 for t [0, T ) and 0 otherwise.

We can write this as

p(t) = rect

t

-

T 2

T

From shift and scaling theorems

P(f ) = Te-jfT sinc(Tf ).

Cuff (Lecture 7)

ELE 301: Signals and Systems

The Derivative Theorem

Fall 2011-12 15 / 37

The Derivative Theorem: Given a signal x(t) that is differentiable almost everywhere with Fourier transform X (f ),

x (t) j2fX (f )

Similarly, if x(t) is n times differentiable, then

d n x (t ) dt n

(j 2f

)nX (f

)

Cuff (Lecture 7)

ELE 301: Signals and Systems

Fall 2011-12 16 / 37

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