Fourier Transform of Signals

-a

)( jIX I

a1

a22

a

)( j<X2

4

4

aa

2

t

)(tx

1

Figure 4.6 Signal of example 4.2 )()( taetx

-a

a

a1

a2)( jx

Figure 4.7

12T

)( jx

1T

1T

t

)(tx

T T t

)( jX

w w

1

W

)(tx

W Wt

Duality Property of Fourier Transform

t2W

)(2 tx2W

2W

)(1 tx

1W

t

1W

1W

)(1jX

1W 1W

1

)(2jX

2W 2W

1

Inverse Relationship between Time & Frequency Domain

Fourier Transforms of Periodic Signals•We can develop Fourier Transform

representations for periodic signals thus allowing us to consider both Periodic and aperiodic signals within a unified context.

• We can construct a FT of a periodic signal directly from its Fourier series representation.

• The resulting transform consists of a train of impulses in the frequency domain with the area of the impulses proportional to FS coefficients.

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• It can be observed that this equation corresponds exactly to the Fourier series representation of a periodic signal as already proved

• Thus the Fourier transform of a periodic signal with Fourier series coefficients can be interpreted as a train of impulses occurring at the harmonically related frequencies

• The area of the impulse at the kth harmonic frequency

is 2 times the kth Fourier series coefficient

Example

kTk

ka

10sin

For the pulse train the Fourier series coefficients already evaluated are:-

Fourier transform of the signal is )0(0sin2)( 1

kk

TkjH

k

The signal is plotted for 14TT

ka

ka0k

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)(jX

22

0 0

k-2 2

T= 4T1

ka

Comparison of )(jX and ka

The only differences are the proportionality factor of 2 and the use of impulses rather than a bar graph

Properties of CT Fourier Transforms

• This property follows directly from the linearity of integrals.

Time-Shifting:

Linearity: If

• Thus, a shift in time does not affect the magnitude of the Fourier transform.

• The effect of a time shift is to introduce a phase shift which is a linear function of the frequency

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•If x(t) is real, then

Conjugation and Conjugate Symmetry

Differentiation

Differentiation in time domain corresponds to multiplication by in frequency domainj

Integration:

Differentiation in time = Multiplication by in frequency domain Integration in time should involve division by iin frequency domain.

)(2jX

W W

1W

W

t

)(2 tX

W

12T

)( jx

1T

1T

)(1tX

t1T 1T

1

F

F

Duality

Parsavel’s RelationThe Parsavel’s relation states that

• The Parsavel’s relation describes that this total energy in a signal can be determined either by computing the energy per unit time and integrating over all time • Or computing the energy per unit frequency and integrating over all the Frequencies • For this reason X(jw) is often referred to as the “energy density spectrum of the signal”

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