Communication Systems

Prof. Chungming Kuo

Chapter 3

Fourier Transform Operations

Fourier Transform Operations When signals are processed in the time domain, various changes occur in their spectra.

It is important to understand how the spectra are affected by these operations.

Fourier Transform Operations (cont.) In this module, the effects of various operations, such as

differentiation and integration, will be studied in terms of the spectral changes.

Fourier Transform Operation Pairs

)(tv )]([)( tvfV F

)()( 21 tbvtav )()( 21 fVbfVa (O-1)

dt

tdv )( )(2 fVfj (O-2)

( )t

v t dt

fj

fV

2)( (O-3)

)( tv )(2 fVe fj (O-4)

)(02 tve tfj )( 0ffV (O-5)

)(atv

a

fV

a

1 (O-6)

Effect of Differentiation on Spectrum

( )V f 2 ( )f V f

f f

dv t dt

j2 fV f (0 – 2)

Effect of Integration on Spectrum

( )V f

f f

( )

2

V f

f

v t dt V f j2 f

t (0 – 3)

Delayed Function

( )v t ( )v t

t t

v t e j 2 f V f (0 – 4)

Effect of Modulation on Spectrum

( )V f 0( )V f f

1f 0 1f f0 1f f0ff f

1f

v t e j 2 f0t V f f0 (0 – 5)

Effect of Time Scaling on Spectrum( )v t

( )v at

( )v at

1a

1a

t

t

t

( )V f

1 fV

a a

1 fV

a a

f

f

f

v at 1a

V fa

(0 – 6)

Spectral Convergence

Time Function Amplitude Rolloff Function has finite 1/f or

discontinuity. -6 dB/octave Slope has 1/f 2 or

finite discontinuity. -12 dB/octave Neither function nor At least 1/f3 or

slope has discontinuity. -18 dB/octave

Summary

Any mathematical operation on a time function results in a change to the spectrum.

Differentiation accentuates high frequencies and diminishes low frequencies.

Integration accentuates low frequencies and diminishes high frequencies.

Summary (cont.)

Multiplication by a complex exponential shifts the spectrum to higher frequencies.

“Speeding up” a signal results in a wider spectrum and vice-versa.