image spectra 2
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Section 29 of Phil's material on image spectra, dated 3.14.13, working two examples with an 11-sample digital signal. Pulse trains w(t) are built from box pulses, with spectrum W(ω) = (1/T1)Xpulse(ω)Y'(ω). With box width T1 the image spectra of Y'(ω) are suppressed in W(ω). With width T1/3 strong image spectra remain. Plots and some equations are lost in the extraction.
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More on image spectra PhL 3.14.13
29. Examples of Image Spectra
Example 1
Consider the following finite length digital signal, which we assume has time spacing T1,
yn (n=-5..5) = { 1,2,3,3,4,1,0.2,-1,-2,-2,-1}
Fig 29.1
All samples other than those shown are 0. Using an arbitrary pulse xpulse, we can construct an amplitude-modulated pulse train w(t) whose spectrum is W(ω) as shown in summary box (25.4),
w(t) = !Syntax Error, I yn xpulse(t -tn) (29.1)
W(ω) = (1/T1)Xpulse(ω) Y'(ω) , (29.2)
where from summary box (23.5),
Y'(ω) ≡ T1!Syntax Error, Iyn e-iωnT . (29.3)
If we were to select xpulse(t) to be a box of height 1 and width T1, then we could regard the stair-step outline function shown in Fig 29.1 as a candidate analog signal y(t) whose samples are the yn. In this special case, y(t) = w(t) so Y(ω) = W(ω). From (9.2) we have Xbox(ω) = T1sinc(ωT1/2) so that using (29.2),
Y(ω) = W(ω) = sinc(ωT1/2) Y'(ω) = T1 sinc(ωT1/2) !Syntax Error, Iyn e-iωnT
Y(ω) is the Fourier Integral Transform spectrum of the stair-step signal in Fig 29.1. Here are some plots
Plot of |Y'(ω)|
Plot of |W(ω)| in red, 15* |sinc(ωT1/2)| in blue
In this example, we see the expected image spectra in Y'(ω) in the first plot, but these spectra are not present in the spectrum W(ω) of the physical signal w(t).
Example 2
We use the same sample values yn as in the previous Example, but the box width is τ = T1/3 instead of
τ =T1. So our analog signal w(t) is now this set of spikes,
Signal w(t) for Example 2.
From (9.2) now have Xbox(ω) = (T1/3) sinc(ω*T1/6), so the spectrum of this w(t) is
W(ω) = (1/T1)[ (T1/3) sinc(ω*T1/6)]Y'(ω) = (1/3) sinc(ω*T1/6)Y'(ω)
with the same Y'(ω) as in the previous example. Here is a plot of the new W(ω),
Plot of |W(ω)| for Example 2 in red, 5 |sinc(ω*T1/6)| in blue.
In this Example, the physical analog signal w(t) does have some strong image spectra in addition to the central main spectrum
Suppose for the corresponding analog function y(t) we take the stair-step outline of the digital signal.
w(t) = y(t) x(t) = !Syntax Error, I yn xpulse(t -tn)
where
xpulse(t ) = 1 [ θ(t + T1/2) - θ(t - T1/2) ] = box of height 1 and width T1