<p> Stem style construct </p>

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Fourier "Period Unit Sampling"

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U 1 - :

L 1 :

Pule train N x f L - x ≤ (x 2 N * ≤ (& U 1 - (k^1 x k L * - sign + 2 / (* (k 1 N sum * NOT accounted if :

Pulse train ... set N = even U = magnitude ... L = width. Generates N/2 complete pulses. Experiment: U := ± 1

0 8 x f 2 1 sys

L period ≡

L 1

λ 0.5 L * :

N 32 :

U 1 :

<= Unit sampling

U L 21 mat magnitude window width 21 mat ≡

a A 1 el 1 U * λ * : n 2 N range q n el U 2 n * π * λ * sin n π * / * eval : for A q stack 31 line :

b n 1 N range B n 1 + el U 1 - 2 n * π * λ * cos + n π * / (* - eval : for B 21 line :

ab a b augment :

x p a 1 el a n 1 + el n π * x * L / cos * b n 1 + el n π * x * L / sin * + (n 1 N sum + :

P x 03 0.01 range : i 1 x rows range u i el x i el : v i el x i el p eval : 21 line for u v augment 31 line :

Usage:

In practical applications, we enter soon the Fourier domain. Once in it, there is no more point going back in the continuous domain for sampling/analysing part of other signal, we can just rescue the project from this technique [just demo]. Intuitively, it introduces the reader to more powerful "Wavelets".

Extract & plot "Unscaled" spectrum

Q n 1 N range c n el a n el i (b n el * - : for c 21 line :

Not used Ω Q :

b 0 0.63662 0 0.212207 0 0.127324 0 0.090946 0 0.070736 0 0.057875 0 0.048971 0 0.042441 0 0.037448 0 0.033506 0 0.030315 0 0.027679 0 0.025465 0 0.023579 0 0.021952 0 0.020536 0331 mat

spectral t 1 N 2 range : i 2 N range v i el Q i el abs : v 1 el a 1 el (: 21 line for t v augment 31 line :

Isolated for stem style spec spectral 2 col :

DC 0 a 1 el 0022 mat :

<= Fake to view DC

<p> Stem style construct </p>

iii 1 N range q iii el 3 : q N el 2 : 21 line for

ii 1 N range v ii el ii : for

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

u q v replicate :

i 1 N range S 3 i * el spec i el : for

V u S augment :

V V 3 V rows 12 submatrix :

Unscaled spectrum. Remember: the 1rst component of a spectrum is the DC component. Past the 2nd component, all odd components are nill. Tabulate "spec" to convince yourelf.

V DC 2 1 sys