Page 257 - Build Your Own Transistor Radios a Hobbyists Guide to High-Performance and Low-Powered Radio Circuits
P. 257
2'П
+ ) J
~
Х 21Т .t + . . . 15-1 '
з N
where N is а positive integer, f s is the sampling frequency in hertz, t is time, and ОС
is the average value of the pulse train. The DC term is also found Ьу taking the
average value of опе period Р of the pulse. And а N i5 а weighting factor based оп
the frequency coefficients of а Fourier series of the pulse-train signal.
We сап define the input signal that wНl Ье 5ampled а5
)
And because the multiplication of the periodic pulse-train signal and the input signal
amounts to sampling of the input signal, we have the sampled input signal as
2 х 211" l +
l ]
.~
Expanding the terms, we get
[Ь\ СО 21Т iJ}l На со 2~ l J +
2
2'П i 1l)][a (Х 1Т [)] .+
[Ь1 (2'П irll Л Х 21ТtfJ)] (15- )
Now let's take а look at some of the product terms:
1. [Ь 1 соs(2пfiпt)]DС is just the input signal that is scaled Ьу the average ОС
voltage.
2. [Ь 1 соs(2пfiпt)][аl соs(2пfst)]is the input sigпаl multiplied Ьу а fundamental
sampling-frequency sinusoidal waveform.
З. [Ь 1 соs(2пfiпt)][а2 cos(2 х 2пf t)] is the input signal multiplied Ьу а
s
second-harmonic sampling-frequency sinusoidal wavefoгm.
4. [Ь 1 соs(2пfiпt)][аз соs(З х 2пf s t)] is the input signal multiplied Ьу а
third-harmonic sampling-frequency sinusoidal waveform.
s
S. [Ь 1 соs(2пfiпt)][аN cos(N х 2пf t)] is the input signal multiplied Ьу ап Mh
harmonic 5ampling-frequency sinusoidal waveform.
50, from the ргесеdiпя, we see for а sampling system that there is а linear term
from the first prod'uct term, as well as multiplication (mixing) between the input
signal and signals of the fundamental and harmonics of the sampling frequency
from the subsequent terms. For 5ampling with а narrow pulse train ог using ап
ideal sampling pulse train, the harmonic signal amplitudes аге equal to that of the
fundamental-frequency signal. That is, with ideal sampling, аl = а2 = ... = aN.
Finite Pulse-Width Signals
For а finite pulse-width 5ignal, calculating the weighting factor5 сап Ье challenging,
50 heгe is the result foг апу Nfoг а pulse centeгed at the oгigin:
