spectral (version 2.0)

BP: Simple bandpass function

Description

This function represents a simple weightening procedure for spectral filtering accoring to the type ("poly", "sinc", "bi-cubic", "gauss") provided.

Usage

BP(f, fc, BW, n = 3, type = "poly")

Arguments

f

vector of frequencies

fc

center frequency

BW

bandwidth, with w[ abs(f - fc) > BW ] == min

n

degree of the polynom, n can be real, e.g. n = 1.6 as sinc alike

type

Type of weightening function: "poly", "sinc", "bi-cubic", "exp", can be abbreviated

Value

This function returns a weight vector [0..1], which is to apply to the frequency vector f in a top level function

Details

The band pass is represented troughout a function in the form of four different types:

1. polynominial function $$w = 1 - |((f - fc) / BW)|^n $$ with the degree \(n\). The parameter \(fc\) controlls the center frequency and desired band width BW. Outside the band width $$|f - fc| > BW$$ the result is forced to zero. With n = 1.6 a quasi sinc-filter without side bands can be constructed. A quasi rectangular window can be gained by setting n > 5.

2. sinc function corresponds to a rectangular observation window in time domain with $$\Delta T ~ 1/BW$$. It values ALL frequencies according to the si(x) function. Calculation speed might be reduced.

3. bi-cubic encounters 2nd order interpolation kernel, providing a quasi rectangular observation window.

4. exponential Gauss curve. Here the band width is defined as the value of 90

Examples

# NOT RUN {
f <- seq(-50,50,by = 1e-2)
fc <- 0.3
BW <- 0.75


par(mfrow = c(2,1))

curve(BP(x,fc = fc, BW = BW, type = "p"), -2,2, ylim = c(-0.2,1)
      ,main = "Filter weights"
      ,xlab = "fx",ylab = "w"
)
curve(BP(x,fc = fc, BW = BW, type = "s"), add = TRUE, lty = 2)
curve(BP(x,fc = fc, BW = BW, type = "b"), add = TRUE, lty = 3)
curve(BP(x,fc = fc, BW = BW, type = "g"), add = TRUE, lty = 4)

abline(v = c(fc,fc+BW,fc-BW), lty = 3, col = "grey")

# the corresponding Fourier-Transforms

ty <- c("p","s","b","g")
A0 <- integrate(BP,fc = fc, BW = BW, type = "s",lower = -2,upper = 2)$value

plot(NA,NA,xlab = "x", ylab = "|A|"
     ,main = "corresponding convolution kernels"
     ,xlim = 2*c(-1,1),ylim = c(0, sqrt(2)*A0/(length(f)*BW*min(diff(f))) )
)
for(i in 1:length(ty))
{
  FT <- spec.fft(y = BP(f,fc,BW,type = ty[i]))
  lines(FT$fx * length(FT$fx) / diff(range(f)),Mod(FT$A),lty = i)

}
# }