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CVT

Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

Sajid Ali

SEECS-NUST

November 20, 2017

DRAFT

CVT

Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Concept

The Fourier series is used to convert a periodic signal (func-tion) in terms of discrete oscillatory modes of different fre-quencies such that all frequencies sum up to the frequencyof original signal. Sometimes we confront with a situation inwhich the signal is not periodic. For example, suppose wehave an electrical circuit in which some switch is off for alltime and turned on for certain time. This is an example of anaperiodic function. Now if we need to use this information insome kind of communication then we need to perform Fourieranalysis.The Fourier transform provides us a way to express aperiodicfunctions in terms of oscillations except that we now requireindefinitely large number of frequencies (continuous) to addup (integrate) to the original function.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Example

In order to see how Fourier transform deal with aperiodic func-tion let us work on the above mentioned problem. Mathemat-ically, the given function is

f (x) =

0, −∞ < x < −L,1, −L < x < L,

0, L < x <∞

This is not periodic. We now consider a similar function whichis periodic fp(x + 2L) = fp(x)

fp(x) =

{1, −L < x < L,

0, otherwise.

Note that if we extend the period such that L→∞, then wenote that

f (x) = limL→∞

fp(x).

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Example

Therefore, aperiodic function f (x) can be thought as a limit-ing case of a periodic function fp(x) as the period of formerbecomes very large. We have obtained the Fourier series of fpin class and analyzed it for different periods 2L = 4, 2L = 8and 2L = 16. We sketched the graphs of Fourier coefficientsbn vs frequency wn = kn = 2πn/2L. It turned out that inlarger periods the frequencies required becomes indefinitelylarge and discrete wn replaces with a continuous variable w .

For details, please refer to the class lecture or sit down anddo it again yourself or discuss with your class fellows or dropby my office with an appointment.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

What if we extend the period L→∞?

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

What if we extend the period L→∞?

Then, we would probably need to add over all possiblefrequencies which mean we need to replace summation overn in the Fourier series from an integral over variablefrequency. It is known as Fourier transform of a function.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

If f (x) is absolutely integrable on the x−axis and piece-wisecontinuous on every finite interval, then the Fouriertransform f̂ (w) of f (x), exist.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

If f (x) is absolutely integrable on the x−axis and piece-wisecontinuous on every finite interval, then the Fouriertransform f̂ (w) of f (x), exist.

F(f (x)) = f̂ (w) =1√2π

∫ ∞−∞

f (x)e−iwx dx

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

If f (x) is absolutely integrable on the x−axis and piece-wisecontinuous on every finite interval, then the Fouriertransform f̂ (w) of f (x), exist.

F(f (x)) = f̂ (w) =1√2π

∫ ∞−∞

f (x)e−iwx dx

Why to call it transform?

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Fourier Transform

If f (x) is absolutely integrable on the x−axis and piece-wisecontinuous on every finite interval, then the Fouriertransform f̂ (w) of f (x), exist.

F(f (x)) = f̂ (w) =1√2π

∫ ∞−∞

f (x)e−iwx dx

Why to call it transform?The basic reason is that one can view the above complexintegral as follows:

(x , f ) =⇒ (w , f̂ )

i.e., a transformation of the variable f (x) into anothervariable f̂ (w).

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Concept

Fourier Transforms

Applications

Properties

Practice

Inverse Fourier Transform

It is also possible to go back into the old variable(w , f̂ )→ (x , f ) using inverse Fourier transform

F−1(f̂ (w)) = f (x) =1√2π

∫ ∞−∞

f̂ (w)e iwx dw

We will study in detail the two domains, their physicalcharacteristics and relevance in the subsequent slides.

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Concept

Fourier Transforms

Applications

Properties

Practice

Examples

Q.1: Findthe Fourier transform of f (x) = 1, |x | < 1 and 0, otherwise?

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Examples

Q.1: Find the Fourier transform of f (x) = 1, |x | < 1 and0, otherwise?

Q.2: F(f (x)) =?, if f (x) = e−ax for x > 0 and 0, otherwise.

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Concept

Fourier Transforms

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Solution

Q.1: Find the Fourier transform of f (x) = 1, |x | < 1 and0, otherwise?Ans.

F(f (x)) = f̂ (w) =1√2π

∫ 1

−1e−iwx dx

=1

−iw√

2π(e−iw − e iw )

=

√2

π

sinw

w, e iw − e−iw = 2i sinw

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Concept

Fourier Transforms

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Solution

Q.1: Find the Fourier transform of f (x) = 1, |x | < 1 and0, otherwise?Geometry: The sketch of f̂ , is drawn in the w−plane alongwith original function for comparison.

F(f (x)) =

√2

π

sinw

w,

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Concept

Fourier Transforms

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Solution

Q.2: F(f (x)) =?, if f (x) = e−ax for x > 0 and 0,otherwise.Ans. Therefore

F(f (x)) =1√2π

∫ ∞0

e−axe−iwx dx

=1√2π

e−(a+iw)x

−(a + iw)|x=∞x=0

=1

(a + iw)

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Concept

Fourier Transforms

Applications

Properties

Practice

Solution

Q.2: F(f (x)) =?, if f (x) = e−ax for x > 0 and 0,otherwise.Geometry: The real and imaginary pars of f̂ , are sketchedbelow on the w−plane.

f̂ (ω) =1

(a + iw)

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Concept

Fourier Transforms

Applications

Properties

Practice

Example

Geometry: The Fourier transform off (x) = 1, −b < x < b and zero otherwise is

F(f (x)) = f̂ (ω) =sin(bw)

w

where we plot its graph for different values of b = 1, 2, 3, 4.

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Concept

Fourier Transforms

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Properties

Practice

Properties:

1. The Fourier transform is a linear operation,

F(af + bg) = aF(f ) + bF(g)

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Fourier Transforms

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Properties:

1. The Fourier transform is a linear operation,

F(af + bg) = aF(f ) + bF(g)

2. Let f (x) be continuous and f (x)→ 0 as |x | → ∞.Furthermore, let f ′(x) be absolutely integrable then

F(f ′(x)) = iwF(f (x))

which is a remarkable property as it helps to convertderivatives into algebraic relation. A key feature in order tosolve differential equations.Remark. It is important to note that not all type ofdifferential equations can be solved via Fourier transforms. Itworks only for linear differential equations whose solutionsexhibit the behavior that f (x)→ 0 as |x | → ∞.

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Fourier Transforms

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Properties:

3. Show that

F(f ′′(x)) = −w2F(f (x))

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Concept

Fourier Transforms

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Properties:

3. Show that

F(f ′′(x)) = −w2F(f (x))

Under what conditions the above statement is true?

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Concept

Fourier Transforms

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Properties:

3. Show that

F(f ′′(x)) = −w2F(f (x))

Under what conditions the above statement is true?It is true when the first derivative meets the samerequirement as of the original function, i.e.,

f ′(x)→ 0 as |x | → ∞,

and f ′′(x), is absolutely integrable which is necessary beforeapplying a Fourier transform.

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Concept

Fourier Transforms

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Properties

Practice

Properties:

5. For an even function f (−x) = f (x), the Fouriertransform is called Fourier cosine transform

Fc(f (x)) = f̂c(w) =

√2

π

∫ ∞0

f (x) coswx dx

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Sajid Ali

Concept

Fourier Transforms

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Properties:

5. For an even function f (−x) = f (x), the Fouriertransform is called Fourier cosine transform

Fc(f (x)) = f̂c(w) =

√2

π

∫ ∞0

f (x) coswx dx

6. For an odd function f (−x) = −f (x), the Fouriertransform is called Fourier sine transform

Fs(f (x)) = f̂s(w) =

√2

π

∫ ∞0

f (x) sinwx dx

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Fourier Transforms

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Properties:

6. ConvolutionThe convolution of two functions f and g is defined by

h(x) = (f ∗ g)(x) =

∫ ∞−∞

f (p)g(x − p)dp

and if we take the Fourier transform then it is just

F(f ∗ g) =√

2πF(f )F(g)

where both f and g are assumed to be piece wise continuous,bounded and absolutely integrable. This is an importantproperty in which the convolution of two functions is equalto the product of their corresponding Fourier transforms.

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Concept

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Properties

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Properties:

7. In order to find the Fourier transform of a shiftedfunction f (x − a), we obtain it as

F(f (x − a)) = e−iawF(f (x)).

which means the magnitude of the transform remain same,only the phase changes.8. For a re-scaled function f (x/a), we obtain its Fouriertransform as

F(f (ax)) =1

|a|f̂ (ω

a)

which indicates that a wider function in the time-domain is anarrow function in the frequency domain.

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Concept

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Properties

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Properties:

9. In order to find the Fourier transform of cos(ω1x)f (x),we obtain it as

F(cos(ω1x)f (x)) =f̂ (ω − ω1) + f̂ (ω + ω1)

2.

which describes how signals can be mixed at differentimmediate frequencies.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Practice Questions:

1. Convert the following signal in time domain intofrequency domain

f (t) =

{t, −1/2 < t < 1/2

0, otherwise

Sketch the real and imaginary parts of f̂ (ω), and make acomparison with the wave form of f (t) = 1,−1 < x < 1 andf (t) = 0, otherwise.2. Find the corresponding wave form of

f (t) =

{1, {−2 < t < −1}

⋃{1 < t < 2}

0, otherwise

Sketch the graph of f̂ (ω).

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Concept

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Practice Questions:

3. Convert the following signal in time domain intofrequency domain

f (t) =1

t2 + a2, a > 0

Sketch the graph of f̂ (ω). Does it exhibit the wavebehavior? If not then what is the physical reason. Does itcontradict your faith that the Fourier transform always givean oscillating behavior?

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Concept

Fourier Transforms

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Properties

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Practice Questions:

4. Find the corresponding wave form of

f (t) =

{1, b < t < c

0, otherwise

Sketch the graph of f̂ (ω), in the cases when both b and care (i) positive (ii) negative (iii) b negative and a positive.Verify the wave form generated obey the correspondingshifts of rectangle.

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Sajid Ali

Concept

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Practice Questions:

5. Find the corresponding wave form of

f (t) =

{e iat , −b < t < b

0, otherwise

Sketch the graph of both functions f (t) and its FT f̂ (ω).What is the input signal look like? Does it have somefrequency? What if the frequency of the input signal isexactly w? What will be your answer and its justification?Discuss the FT in the cases (i) a = −ω (ii) a = ω/2 (iii)a = ω/3. Using the answer how can we determine the FT of(i) sin(ωt) (ii) sin(2ωt) (iii) cos(ωt) (iv) cos(2ωt), in a finiteinterval and zero elsewhere. Analyze your answer in similarconsiderations.

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Sajid Ali

Concept

Fourier Transforms

Applications

Properties

Practice

Practice Questions:

6. The Fourier transform of a Gaussian function

f (t) = e−at2, a > 0

is the function f̂ (ω) = (1/√

2a)e−ω2/4a. What does the

function represent in the frequency domain? What is sospecial about the value a = 1/2? Identify the changes ingraphs in the ranges i) a < 1/2 and (ii) a > 1/2.

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Sajid Ali

Concept

Fourier Transforms

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Properties

Practice

Practice Questions:

6. Find the corresponding wave form of the triangularfunction

f (t) =

{1− |t|, |t| ≤ 1,

0, otherwise

Sketch the graph of resulting function and see if it agreeswith the graph in time domain.