11.1. The signal becomes separate accounts

Content and key concepts

Re-use of coefficients and parallel amounts are understandable.
Measurement at equal intervals preserves only part of the data. Different background variations sometimes give the same numbers.
A small window passes through a signal or image. Each exit consists of the numbers of the window and the overall coefficients. The frequency record describes the same data differently.
Для изменения рисунка включитеJavaScript; расчёт и теория доступны далее.
From function to sequence
The continuous signal s(t) and the array of x[n] accounts are different. With a uniform sumpling x[N]=s(n/f_s), where f_s is the number of accounts per second and n is the index. The amplitude quantizes separately: even with accurate time, the final number format is rounded.
For frequency cosine f:
x
[
n
]
=
cos
⁡
(
2
π
f
n
/
f
s
)
.
x[n]=\cos(2\pi fn/f_s).
x
[
n
]
=
cos
(
2
π
f
n
/
f
s
​
)
.
At f=3Hz and f_s=16Hz the counts are consistent with cosine 13Hz:13=16–3 and coss(2·n–__)=cosine of theta. This is a verified alias, not an error of calculation. One of these accounts cannot determine which of the two continuous frequencies was reference.
The perfect restoration of the restricted band requires a sampling frequency above the double maximum frequency and a corresponding recovery. The real systems use anti-alias filtering to discreteize. Equality at the border itself requires caution with phase: sinus at f_s/2 can give one zero.
Experiment.
Compare 3 and 13Hz at f_s=16, then zoom in f_S. In manual JSON-massor mode, the reference continuous function is unknown, so the lab does not draw a fake "true wave." Generator controllers in this mode are disabled.
The length of the N recording affects the observation time of N/f_s and the step of the frequency bins f_s/N. More samples does not correct the frequency mix that has already occurred if the f_S itself remains insufficient. This difference will be useful when reading the spectrum.
from math import cos,pi
a=[cos(2*pi*3*n/16) for n in range(16)]
b=[cos(2*pi*13*n/16) for n in range(16)]
assert max(abs(x-y) for x,y in zip(a,b))<1e-12
The code repeats the operations just taken care of.
Completely checked Python-example Chapter
Available with a separate file; the short piece shown does not replace its implementation.
Coss(2pn-ta) is cosine of theta. According to the calculations, these two generators are indistinguishable.
The window moves on the signal. How does this question continue with the result?
Python examples of Chapter 11
Academic base