THE FOURIER TRANSFORM

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CHAPTER 17:

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17.1 The Derivation of the Fourier Transform 17.2 The Convergence of the Fourier Integral

17.3 Using Laplace Transforms to Find Fourier Transforms 17.4 Fourier Transforms in the Limit

17.5 Some Mathematical Properties 17.6 Operational Transforms

17.7 Circuit Applications 17.8 Parseval’s Theorem

Electronic Circuits, Tenth Edition J

ames W. Nilsson | Susan A. Riedel 2

17.1 The Derivation of the Fourier Transform

• The Fourier transform gives a frequency-domain description of an aperiodic time-domain function.

Fourier series

Fourier transform

Allowing the fundamental period T to increase without limit

accomplishes the transition from a periodic to an aperiodic function.

In other words, if T becomes infinite, the function never repeats itself and hence is aperiodic.

Where

1 ^⁄

⁄

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As

T 

^,the transition from a periodic to an aperiodic function

the incremental separation

 

approaches a differential separation

d 

As the period increases, the frequency moves from being a discrete variable to becoming a continuous variable,

As the period increases, the Fourier coefficients get smaller.

However, the limiting value of the product C_nT

17.1 The Derivation of the Fourier Transform

1 1 →

2 a → ∞

→ → ∞ .

→ → ∞

lim , → → ∞

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: Inverse Fourier transform

By multiplying and dividing by T

Transforming frequency domain expression F() into time-domain expression f(t) Fourier transform

Transforming time-domain expression f(t) into frequency domain expression F()

17.1 The Derivation of the Fourier Transform

1 2

→ ∞, s in , →

1

→ , 1⁄ → ⁄2

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A voltage pulse.

• As the time-domain function goes from periodic to aperiodic,

the amplitude spectrum goes from a discrete line spectrum to a continuous spectrum

• The physical interpretation of the Fourier transform V(ω) is therefore a measure of the frequency content of v(t).

Deriving the Fourier transform of the pulse

17.1 The Derivation of the Fourier Transform

⁄

| ^⁄_⁄

2 2

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Transition of the amplitude spectrum as ƒ(t) goes from periodic to aperiodic.

In the form of (sin x)/x by multiplying the numerator and denominator by τ

the expression for the Fourier coefficients

As the time-domain function goes from periodic to aperiodic, the amplitude spectrum goes

from a discrete line spectrum to a continuous spectrum.

17.1 The Derivation of the Fourier Transform

sin ⁄2

⁄2

sin ⁄2

⁄2

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• Depending on the nature of the time-domain signal,

one of three approaches to finding its Fourier transform may be used:

(1)If the time-domain signal is a well-behaved pulse of finite duration, the integral that defines the Fourier transform is used.

(2) If the one-sided Laplace transform of f(t) exists and all the poles of F(s) lies in the left half of the s plane, F(s) may be used to find F(ω).

(3) If f(t) is a constant, a signum function, a step function, or a sinusoidal function, the Fourier transform is found by using a limit process.

17.1 The Derivation of the Fourier Transform

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• A single-valued function that is nonzero over an infinite interval has a Fourier transform if the integral

exists and if any discontinuities in f(t) are finite.

An example:

The decaying exponential function

17.1 The Derivation of the Fourier Transform

|

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The Fourier transform of f(t)

17.1 The Derivation of the Fourier Transform

| 0 1

, 0.

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17.2 The Convergence of the Fourier Integral

• We find the Fourier transform of the approximating function and then evaluate the limiting value of F(ω); finding the F.T of a constant

Finding the Fourier transform of a constant.

Approximating a constant with the exponential function

The approximation of a constant with an exponential function.

, 0

→ 0. → .

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Fourier transform of f(t)

The function generates an impulse function at  = 0 as   0 (1) F() approaches infinity at  = 0 as   0 ; (2) the

width of F() approaches zero as   0 ; and (3) the area under F() is independent of  . The area under F() is the strength of the impulse

In the limit, f(t) approaches a constant A, and F() approaches an impulse function 2A(t)

17.2 The Convergence of the Fourier Integral

F

2

4 2

A 2π

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The reflection of a negative- time function over to the positive-time domain.

17.3 Using Laplace Transforms to Find Fourier Transforms

• Use a table of unilateral, or one-sided, Laplace transform pairs to find the Fourier transform

The Fourier integral converges when all the poles of F(s) lie in the left half of the s plane.

Note that if F(s) has poles in the right half of the

s plane or along the imaginary axis, f(t) does not satisfy the constraint

that | exists

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• The following rules apply to the use of Laplace transforms

1. If f(t) is zero for t < 0^-, we obtain the Fourier transform of from the Laplace transform of simply by replacing s by j

For example,

17.3 Using Laplace Transforms to Find Fourier Transforms

0, 0 ;

cos 0 .

|

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2. Because the range of integration on the Fourier integral goes from -

to , the Fourier transform of a negative-time function exists. A negative- time function is nonzero for negative values of time and zero for positive values of time. To find the Fourier transform of such a function, we

proceed as follows.

First, we reflect the negative-time function over to the positive-time domain and then find its one-sided Laplace transform. We obtain the Fourier transform of the original time function by replacing s with - j . Therefore, when f(t) = 0 for t > 0⁺,

then For example,

17.3 Using Laplace Transforms to Find Fourier Transforms

0, 0 ;

cos 0 .

0, 0 ;

cos 0 .

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The reflection of a negative-time function over to the positive-time domain

The Fourier transform of f(t)

17.3 Using Laplace Transforms to Find Fourier Transforms

|

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3. Functions that are nonzero over all time can be resolved into positive and negative-time functions.

The Fourier transform of the original function is the sum of the two transforms

then

17.3 Using Laplace Transforms to Find Fourier Transforms

0 0

t

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An example

17.3 Using Laplace Transforms to Find Fourier Transforms

1

1 1

2

2 2

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If is even,

If is odd,

17.3 Using Laplace Transforms to Find Fourier Transforms

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17.4 Fourier Transforms in the Limit

The Fourier Transform of a Signum(Sign) Function

The signum function.

Creating a function that approaches the signum function in the limit

sgn

sgn lim

→ , 0.

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A function that approaches sgn(t) as  approaches zero

Because is odd,

17.4 Fourier Transforms in the Limit

1 1

2

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17.4 Fourier Transforms in the Limit

→ 0, → , → 2/

sgn

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17.4 Fourier Transforms in the Limit

The Fourier Transform of a Unit Step Function

The Fourier Transform of a Cosine Function 1

2

1 2sgn 1

2

1

2sgn

2π

1

2 2π

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Using the sifting property of the impulse function

To find the Fourier transform of cos ₀^t

17.4 Fourier Transforms in the Limit

. 2π

cos 2

cos 1

2 1

2 2π 2π

π π

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17.4 Fourier Transforms in the Limit

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17.4 Fourier Transforms in the Limit

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17.5 Some Mathematical Properties

* is a complex quantity and can be expressed in either rectangular or polar

form.

Now we let

sin

sin 17.36

cos

sin

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• The real part of  —that is, —is an even function of ; in other words,

• The imaginary part of F() —that is, B() —is an odd function of ; in other words,

• The magnitude of  —that is, ² ² —is an even function of  ^.

• The phase angle of  —that is,   / —is an odd function of  ^.

• Replacing  ^{by -}  generates the conjugate of F(⁾ ; in other words, ∗

F() properties

17.5 Some Mathematical Properties

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If is an even function,  is real, and if is an odd function,  is imaginary

If is an even function

If is an odd function

17.5 Some Mathematical Properties

2 cos

0

2 sin

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If is an even function, its Fourier transform is an even function,

and if is an odd function, its Fourier transform is an odd function.

If is an even function, from the inverse Fourier integral

*The waveforms of and become interchangeable

if is an even function

17.5 Some Mathematical Properties

1 2

1

2 sin

1

2 cos 0

2

2 cos 2 cos

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17.6 Operational Transforms

• Multiplication

• Addition

• Differentiation

• Integration

• Scale Change

• Translation in the T

• Translation in the F

• Modulation

• Convolution

• Convolution in the F

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17.8 Parseval’s Theorem

Parseval’s theorem:

• The magnitude of the Fourier transform squared is a measure of the e nergy

density (joules per hertz) in the frequency domain.

• Relating the energy associated with a time-domain function of finite energy to the Fourier transform of the function.

Assumption : the time-domain function is either the voltage across or the current in a resistor 1 

The energy associated with this function

Parseval’s theorem holds that this same energy can be calculated by an integration in the frequency domain,

1 2

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The average power associated with time- domain signals of finite energy is zero when averaged over all time.

Note that  ∗ 

Note that F(-⁾ ^F*() = the magnitude of F(^)squared

17.8 Parseval’s Theorem

1 2 1

2

1

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A Demonstration of Parseval’s Theorem If

The Fourier transform of

17.8 Parseval’s Theorem

| |

2 2

1 2

2 2

1 4 4 1

2

1tan 2 0

2 0 0 1

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The Interpretation of Parseval’s Theorem

: the magnitude of the Fourier transform squared, | | , is an energy density (in joules per hertz).

where  2 2 df is the energy in an infinitesimal band of frequencies (df ), and the total 1  energy associated with is the summation (integration) of  2 2 df over all

frequencies

17.8 Parseval’s Theorem

1 | 2 | 2 2 | 2 | ,

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• The Fourier transform permits us to associate a fraction of the total energy contained in with a specified band of frequencies.

The graphic interpretation

17.8 Parseval’s Theorem

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The Energy Contained in a Rectangular Voltage Pulse

The rectangular voltage pulse and its Fourier transform.

(a) The rectangular voltage pulse. (b) The Fourier transform of v(t).

Calculating the energy associated with a rectangular voltage pulse

*when time is compressed, frequency is stretched out and vice versa

17.8 Parseval’s Theorem

sin /2 /2

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Parseval’s theorem to calculate the fraction of the total energy associated with that lies in the frequency range 0  2/

we let

Integrating the integral

17.8 Parseval’s Theorem

1 ^/ sin /2

/2

2

| 1

sin 2

0 sin 2

2 ² ²

2

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The total 1  energy associated with v(t) can be calculated either from

the time-domain integration or the evaluation with the upper limit equal to infinity.

the total energy

The fraction of the total energy associated with the band of frequencies between 0 and 2/

17.8 Parseval’s Theorem

(1.41815)

2

2 ² 1.41815

2

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THE FOURIER TRANSFORM - KOCw

CHAPTER 17: