Function call

fourierSeriesSampledReal(t,u,T,hMax)

Description

Calculates the real Fourier coefficients of a sampled function u(t) based on the data pairs t and u

Function arguments

t Vector of sample times

u Data vector of which Fourier coefficients shall be determined of, i.e., u[k] is the sampled data value associated with t[k]

T Time period of u

hMax Maximum harmonic number to be determined

Return values

h Vector of harmonic numbers start at, i.e., h[1] = 0 (dc value of u), h[hMax+1] = hMax

f Frequency vector associated with harmonic numbers, i.e. f = h / T

a Cosine coefficients of Fourier series, where a[1] = dc value of u

b Sine coefficients of Fourier series, where b[1] = 0

Examples

Copy and paste code:

# Fourier approximation of square wave form
ts = [ 0; 1; 2; 3; 4; 5; 6; 7; 8; 9]    # Sampled time data
us = [-1;-1;-1;-1;-1; 1; 1; 1; 1; 1]    # Step function of square wave
T = 10                      # Period
hMax = 7                    # Maximum harmonic number
(h, f, a, b) = fourierSeriesSampledReal(ts, us, hMax)
(t, u) = fourierSeriesSynthesisReal(f, a, b)
using PyPlot
# Extend (t, u) by one element to right periodically
(tx, ux) = repeatPeriodically(ts, us, right=1)
plot(tx, ux, marker="o")    # Square wave function
plot(t, u)                  # Fourier approximation up to hMax = 7

Function call

fourierSeriesStepReal(t, u, T, hMax)

Description

Calculates the real Fourier coefficients of a step function u(t) based on the data pairs t and u according to the following variables:

\[\mathtt{a[k]} = \frac{2}{\mathtt{T}} \int_{\mathtt{t[1]}}^{\mathtt{t[1]+T}} \mathtt{u(t)} \cos(k\omega \mathtt{t}) \operatorname{d}\mathtt{t} \\\]

\[\mathtt{b[k]} = \frac{2}{\mathtt{T}} \int_{\mathtt{t[1]}}^{\mathtt{t[1]+T}} \mathtt{u(t)} \sin(k\omega \mathtt{t}) \operatorname{d}\mathtt{t}\]

Function arguments

t Time vector instances where a step occurs

u Data vector of which Fourier coefficients shall be determined of, i.e., u[k] is the data value from the right at t[k]

T Time period of u

hMax Maximum harmonic number to be determined

Return values

h Vector of harmonic numbers start at, i.e., h[1] = 0 (dc value of u), h[hMax+1] = hMax

f Frequency vector associated with harmonic numbers, i.e. f = h / T

a Cosine coefficients of Fourier series, where a[1] = dc value of u

b Sine coefficients of Fourier series, where b[1] = 0

Examples

Copy and paste code:

# Fourier approximation of square wave form
ts = [0; 1]                 # Sampled time data
us = [-1; 1]                # Step function of square wave
T = 2                       # Period
hMax = 7                    # Maximum harmonic number
(h, f, a, b) = fourierSeriesStepReal(ts, us, T, hMax)
(t, u) = fourierSeriesSynthesisReal(f, a, b)
using PyPlot
# Extend (t, u) by one element to right periodically
(tx, ux) = repeatPeriodically(ts, us, right=1)
step(tx, ux, where="post")  # Square wave function
plot(t, u)                  # Fourier approximation up to hMax = 7