Information Technology Reference
In-Depth Information
fundamental. The shape of the spectrum is a sin
x
/
x
curve. If a square wave has a sin
x
/
x
spectrum, it follows that a
filter with a rectangular impulse response will have a sin
x
/
x
spectrum.
Figure 3.35:
Fourier analysis of a square wave into fundamental and harmonics.
A
, amplitude; , phase of
fundamental wave in degrees; 1, first harmonic (fundamental); 2, odd harmonics 3-15; 3, sum of harmonics 1-15;
4, ideal square wave.
A low-pass filter has a rectangular spectrum, and this has a sin
x
/
x
impulse response. These characteristics are
known as a transform pair. In transform pairs, if one domain has one shape of the pair, the other domain will have
the other shape.
Figure 3.36
shows a number of transform pairs.
Figure 3.36:
Transform pairs. At (a) the dual of a rectangle is a sin
x
/
x
function. If one is time domain, the other is
frequency domain. At (b), narrowing one domain widens the other. The limiting case of this is (c). Transform of the
sin
x
/
x
squared function is triangular (d).
At (a) a square wave has a sin
x
/
x
spectrum and a sin
x
/
x
impulse has a square spectrum. In general the product of
equivalent parameters on either side of a transform remains constant, so that if one increases, the other must fall. If
(a) shows a filter with a wider bandwidth, having a narrow impulse response, then (b) shows a filter of narrower
bandwidth which has a wide impulse response. This is duality in action. The limiting case of this behaviour is where
one parameter becomes zero, the other goes to infinity. At (c) a time-domain pulse of infinitely short duration has a
flat spectrum. Thus a flat waveform, i.e. DC, has only zero in its spectrum. The impulse response of the optics of a
laser disk (d) has a sin
2
x
/
x
2
intensity function, and this is responsible for the triangular falling frequency response of
the pickup. The lens is a rectangular aperture, but as there is no such thing as negative light, a sin
x
/
x
impulse
response is impossible. The squaring process is consistent with a positive- only impulse response. Interestingly the
transform of a Gaussian response in still Gaussian.
