Signals and Systems - Electrical Engineering

344 CHAPTER 5: Frequency Analysis: The Fourier Transform

If the input of the spectrum analyzer isx(t), the output of either the fixed- or the adjustable-bandpass

filters in the implementations—assumed to have a very narrow bandwidth1—would be

y(t)=

1

2 π

 0 +∫0.51

 0 −0.51

X()ejtd

≈

1

2 π

1X( 0 )ej^0 t

Computing the mean square of this signal we get

1

T

∫

T

|y(t)|^2 dt=

(

1

2 π

) 2

|X( 0 )|^2

which is proportional to the power or the energy of the signal in 0 ±1. A similar computation

can be done at each of the frequencies of the input signal.

Remarks

n The bank-of-filter spectrum analyzer is used for the audio range of the spectrum.

n Radio frequency spectrum analyzers resemble an AM demodulator. It usually consists of a single narrow-

band intermediate frequency (IF) bandpass filter fed by a mixer. The local oscillator sweeps across the

desired band, and the power at the output of the filter is computed and displayed on a monitor.

5.8 Additional Properties

We consider now some additional properties of the Fourier transform, some of which look like those

of the Laplace transform whens=jand some are different.

5.8.1 Time Shifting

Ifx(t)has a Fourier transformX(), then

x(t−t 0 ) ⇔ X()e−jt^0

x(t+t 0 ) ⇔ X()ejt^0 (5.26)

The Fourier transform ofx(t−t 0 )is

F[x(t−t 0 )]=

∫∞

−∞

x(t−t 0 )e−jtdt

=

∫∞

−∞

x(τ)e−j(τ+t^0 )dτ=e−jt^0 X()

where we changed the variable toτ=t−t 0. Likewise forx(t+t 0 ).