11.4. Same data at frequencies

Content and key concepts

On the previous page, which changes without a window twist.
Set the signal to the variable components. The frequencies do not add new measurements, but provide another way to describe the former ones.
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.
The same operation in another submission
DFT converts National accounts to the integrated coefficients:
X
[
k
]
=
∑
n
=
0
N
−
1
x
[
n
]
e
−
2
π
i
k
n
/
N
.
X[k]=\sum_{n=0}^{N-1}x[n]e^{-2\pi i kn/N}.
X
[
k
]
=
n
=
0
∑
N
−
1
​
x
[
n
]
e
−
2
πikn
/
N
.
In the direct transformation of this chapter there is no multiplier1/N; it is used in the reverse. To display magnitude, we show separately X[k]/N. H08 captures the sign, the norm and the order of positive/negative frequancy bins.
DFT, not FFT
.
For x=[1.0, -1.0] the coefficients are equal[.2.0.2]. For the period 4, the main non-zero pair corresponds to the frequency of the 1/4 cycle. The number2 in the spectrum does not mean the amplitude2 of the reference cosine: the positive and negative parts divide the contribution.
For line control, it's true Y=XH if both sequences are supplemented by zeros at least to N+K-1. Without sufficient addition, the DFT product describes circular convolution where the tail is wrapped. The laboratory calculates both sides of the identity independently and shows residual.
Experiment.
For h=[5.5], the frequency response H(
from cmath import exp
from math import pi
x=[1,0,-1,0]
X=[sum(v*exp(-2j*pi*k*n/4) for n,v in enumerate(x)) for k in range(4)]
assert max(abs(a-b) for a,b in zip(X,[0,2,0,2]))<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.
Without L≥N+K-1 a circular convolution with wrapping is produced.
Two-dimensional image window. How does this question continue the result?
Python examples of Chapter 11
Academic base