1. Introduction
Most engineers use Matlab (or its open source alternative Octave) to
solve their signal processing problems. Languages such as R or Python
are not immediately thought of for signal processing, although this has
changed a bit in the last few years with the development of the R signal package
and Python scipy.signal.
The package gsignal aims to further stimulate the use of
R for signal processing tasks. It is ported from the Octave signal package, version
1.4.1 (2019-02-08). The package contains a variety of signal processing
tools, such as signal generation and measurement, correlation and
convolution, filtering, FIR and IIR filter design, filter analysis and
conversion, power spectrum analysis, system identification, decimation
and sample rate change, and windowing.
This vignette provides a brief and general overview of some of
gsignal’s functions.
library(gsignal)
#>
#> Attaching package: 'gsignal'
#> The following objects are masked from 'package:stats':
#>
#> filter, gaussian, poly
2. Signal generation and measurement
The function pulstran can be used to generate trains of
pulses based on samples of a continuous function (which can be
user-defined). The following figures show a periodic rectangular pulse,
an asymmetric sawtooth pulse, a periodic Gaussian waveform, and a custom
pulse train.
op <- par(mfrow = c(2, 2))
## periodic rectangular pulse
t <- seq(0, 60, 1/1e3)
d <- cbind(seq(0, 60, 2), sin(2 * pi * 0.05 * seq(0, 60, 2)))
y <- pulstran(t, d, 'rectpuls')
plot(t, y, type = "l", xlab = "", ylab = "",
main = "Periodic rectangular pulse")
## assymetric sawtooth waveform
fs <- 1e3
t <- seq(0, 1, 1/fs)
d <- seq(0, 1, 1/3)
x <- tripuls(t, 0.2, -1)
y <- pulstran(t, d, x, fs)
plot(t, y, type = "l", xlab = "", ylab = "",
main = "Asymmetric sawtooth ")
## Periodic Gaussian waveform
fs <- 1e7
tc <- 0.00025
t <- seq(-tc, tc, 1/fs)
x <- gauspuls(t, 10e3, 0.5)
ts <- seq(0, 0.025, 1/50e3)
d <- cbind(seq(0, 0.025, 1/1e3), sin(2 * pi * 0.1 * (0:25)))
y <- pulstran(ts, d, x, fs)
plot(ts, y, type = "l", xlab = "", ylab = "",
main = "Gaussian pulse")
## Custom pulse trains
fnx <- function(x, fn) sin(2 * pi * fn * x) * exp(-fn * abs(x))
ffs <- 1000
tp <- seq(0, 1, 1 / ffs)
pp <- fnx(tp, 30)
fs <- 2e3
t <- seq(0, 1.2, 1 / fs)
d <- seq(0, 1, 1/3)
dd <- cbind(d, 4^-d)
z <- pulstran(t, dd, pp, ffs)
plot(t, z, type = "l", xlab = "", ylab = "",
main = "Custom pulse")
A number of waveform generating functions are available, such as
chirp, cmorwavf, diric,
gauspuls, gmonopuls, mexihat,
meyeraux, morlet, rectpuls,
sawtooth, square, and
tripuls.
The function findpeaks can be used to determine (local)
minima and maxima in a signal, as the following figures show.
t <- 2 * pi * seq(0, 1,length = 1024)
y <- sin(3.14 * t) + 0.5 * cos(6.09 * t) +
0.1 * sin(10.11 * t + 1 / 6) + 0.1 * sin(15.3 * t + 1 / 3)
data1 <- abs(y) # Positive values
peaks1 <- findpeaks(data1)
data2 <- y # Double-sided
peaks2 <- findpeaks(data2, DoubleSided = TRUE)
peaks3 <- findpeaks (data2, DoubleSided = TRUE, MinPeakHeight = 0.5)
op <- par(mfrow=c(1,2))
plot(t, data1, type="l", xlab="", ylab="")
points(t[peaks1$loc], peaks1$pks, col = "red", pch = 1)
plot(t, data2, type = "l", xlab = "", ylab = "")
points(t[peaks2$loc], peaks2$pks, col = "red", pch = 1)
points(t[peaks3$loc], peaks3$pks, col = "red", pch = 4)
par (op)
title("Finding the peaks of smooth data is not a big deal")
t <- 2 * pi * seq(0, 1, length = 1024)
y <- sin(3.14 * t) + 0.5 * cos(6.09 * t) + 0.1 *
sin(10.11 * t + 1 / 6) + 0.1 * sin(15.3 * t + 1 / 3)
data <- abs(y + 0.1*rnorm(length(y),1)) # Positive values + noise
peaks1 <- findpeaks(data, MinPeakHeight=1)
dt <- t[2]-t[1]
peaks2 <- findpeaks(data, MinPeakHeight=1, MinPeakDistance=round(0.5/dt))
op <- par(mfrow=c(1,2))
plot(t, data, type="l", xlab="", ylab="")
points (t[peaks1$loc],peaks1$pks,col="red", pch=1)
plot(t, data, type="l", xlab="", ylab="")
points (t[peaks2$loc],peaks2$pks,col="red", pch=1)
par (op)
title(paste("Noisy data may need tuning of the parameters.\n",
"In the 2nd example, MinPeakDistance is used\n",
"as a smoother of the peaks"))
3. Filter Design
The gsignal package contains functions for designing
lowpass, highpass, bandpass, and bandstop filters. Both Finite Impulse
Response (FIR) and Infinite Impulse Response (IIR) filters can be
designed. The freqz function displays the frequency
response’s magnitude and phase of the filter.
FIR filters
A FIR filter is a filter whose impulse response settles to zero in
finite time. This is in contrast to IIR filters, which have internal
feedback causing them to have an infinitely long impulse response
(although usually decaying).
For causal discrete-time FIR filters the output is a weighted sum of
the most recent input values. Compared to IIR filters, advantages of FIR
filters are they are inherently stable (because there is no feedback
that propagates indefinitely), and that they have linear phase (constant
across frequencies). The main disadvantage is that they require more
computation time to obtain sharp transition bands.
The package gsignal contains various methods to design
digital FIR filters. The functions fir1, fir2,
and kaiserord use the windowing
method, in which a window is applied to the truncated inverse
Fourier transform of the filter’s frequency response. The function
firls is an extension of the fir1 and
fir2 functions that uses a least-squares approach to
minimize errors between the specified and the actual frequency response
over sub-bands of the frequency range. The Parks-McClellan
method using the Remez exchange
algorithm for finding an optimal equiripple set of filter
coefficients is used by the remez function. The
cl2bp function allows designing FIR filters without
explicitly defining the transition bands for the magnitude response.
Below are some examples of FIR filter design. The magnitude and the
phase of the filter’s frequency response are plotted by the function
freqz.
## FIR filter design by windowing
# low-pass filter 10 Hz
fs = 256
h <- fir1(40, 10/ (fs / 2), "low")
freqz(h, fs = fs)
# observe the effect of filter length
h <- fir1(80, 10/ (fs / 2), "low")
freqz(h, fs = fs)
# fir2 allows specifying arbitrary frequency responses
f <- c(0, 0.3, 0.3, 0.6, 0.6, 1)
m <- c(0, 0, 1, 1/2, 0, 0)
fh <- freqz(fir2(100, f, m))
op <- par(mfrow = c(1, 2))
plot(f, m, type = "b", ylab = "magnitude", xlab = "Frequency")
lines(fh$w / pi, abs(fh$h), col = "blue")
legend("topright", legend = c("specified", "actual"), lty = 1,
pch = c(1, NA), col = c("black", "blue"))
# plot in dB:
plot(f, 20*log10(m+1e-5), type = "b", ylab = "dB", xlab = "Frequency")
lines(fh$w / pi, 20*log10(abs(fh$h)), col = "blue")
par(op)
title("specify arbitrary frequency responses with fir2")
## 50 Hz notch filter with remez
fs <- 200
nyquist <- fs / 2
f <- c(0, 48.5 / nyquist, 49.5 / nyquist, 50.5 / nyquist, 51.5 / nyquist, 1)
a <- c(1, 1, 0, 0, 1, 1)
h <- remez(200, f, a)
freqz(h, fs = fs)
Compensating for filter delay in FIR filters
Filtering causes a delay because weighted samples in the past are
used. FIR filters have a linear phase, so in the time domain this delay
is constant, namely \(N / 2\), where
\(N\) is the filter length (or ‘number
of taps’). The function grpdelay can be used to calculate
the group
delay; see Figure (a) below. Because phase is linear, it is easy to
to compensate for the filter delay as shown in Figure (b) below.
op <- par(mfrow = c(2, 1))
# design the filter
fs = 256
h <- fir1(40, 30/ (fs / 2), "low")
# group delay is constant at N/2
gd <- grpdelay(h)
plot(gd, ylim = c(0, 40),
main = paste("(a) Group delay for FIR filters is constant\n",
"(here 40 / 2 = 20)"))
# filter electrocardiogram data with added noise
data(signals)
npts <- nrow(signals)
ecg <- signals$ecg + 1000 * runif(npts)
time <- seq(0, 10, length.out = npts)
plot(time, ecg, type = "l", main = "(b) Example ECG signal",
xlab = "Time", ylab = "", xlim = c(0,2))
title(ylab = expression(paste("Amplitude (", mu, "V)")), line = 2)
f1 <- filter(h, ecg)
lines(time, f1, col = "red", lwd = 2)
delay <- mean(gd$gd)
f2 <- c(f1[(delay + 1):npts], rep(NA, delay + 1))
lines(time, f2, col = "blue", lwd = 2)
legend("topright", legend = c("Original", "Filtered", "Corrected"),
lty = 1, lwd = c(1, 2, 2), col = c("black", "red", "blue"))
IIR filters
Infinite Impulse Response, or recursive, filters are an efficient way
of achieving a long impulse response by not only using past input
samples, but also past output samples. Hence, an element of feedback
(recursion) is used. IIR filters are specified by a set of
feedback coefficients (usually termed \(a\)), in addition to feedforward
coefficients (\(b\)) as used in FIR
filters.
Advantages of IIR filters compared to FIR filters are related to
their efficiency in implementation. IIR filters usually require (much)
fewer filter coefficients, implying a correspondingly fewer number of
calculations. On the other hand, the impulse response of IIR filters
does not always decay to zero, which may result in filter instability
(see the example below). In addition, the phase of IIR filters is not
linear but frequency dependent. Forward and reverse filtering
(filtfilt) results in zero phase at the expense of
additional computing time (there is no free lunch).
Some important types of IIR filters are:
- Butterworth filters have frequency response that is as flat as
possible in the passband (function
butter);
- Chebyshev filters are IIR filters having a steeper roll-off than
Butterworth filters, and either have a passband ripple (Type I -
function
cheby1), or a stopband ripple (Type II - function
cheby2);
- Elliptic filters with equalized ripple (equiripple) behavior in both
the passband and the stopband (function
ellip);
- (Analog) Bessel filters with a maximally linear phase response.
The following figure compare the frequency responses of (a) 5th order
Butterworth and Chebyshev filters, (b) 5th order Butterworth and
elliptic filters, and (c) type I and type II Chebyshev filters.
op <- par(mfrow = c(3,1))
# compare Butterworth and Chebyshev filters.
bfr <- freqz(butter(5, 0.1))
cfr <- freqz(cheby1(5, .5, 0.1))
plot(bfr$w, 20 * log10(abs(bfr$h)), type = "l", ylim = c(-80, 0),
xlab = "Frequency (rad)", ylab = c("dB"),
main = "(a) Butterworth and Chebyshev")
lines(cfr$w, 20 * log10(abs(cfr$h)), col = "red")
legend("topright", legend = c("5th order Butterworth", "5th order Chebyshev"),
lty = 1, col = c("black", "red"))
# compare Butterworth and elliptic filters.
efr <- freqz(ellip(5, 3, 40, 0.1))
plot(bfr$w, 20 * log10(abs(bfr$h)), type = "l", ylim = c(-80, 0),
xlab = "Frequency (rad)", ylab = c("dB"),
main = "(b) Butterworth and Elliptic")
lines(efr$w, 20 * log10(abs(efr$h)), col = "red")
legend ("topright", legend = c("5th order Butterworh", "5th order Elliptic"),
lty = 1, col = c("black", "red"))
# compare type I and type II Chebyshev filters.
c1fr <- freqz(cheby1(5, .5, 0.1))
c2fr <- freqz(cheby2(5, 20, 0.1))
plot(c1fr$w, 20 * log10(abs(c1fr$h)), type = "l", ylim = c(-80, 0),
xlab = "Frequency (rad)", ylab = c("dB"),
main = "(c) Type I and II Chebyshev")
lines(c2fr$w, 20 * log10(abs(c2fr$h)), col = "red")
legend ("topright", legend = c("5th order Type I", "5th order Type II"),
lty = 1, col = c("black", "red"))
Numerical precision and stability
Using IIR filter coefficients \(b,
a\) can cause numerical problems. Therefore, IIR filter design
functions in gsignal have an output parameter,
allowing the filter coefficients to be returned in one of three
forms:
- Arma, a
list containing the moving average polynomial
(feedforward) coefficients \(b\), and
the autoregressive (recursive, feedback) coefficients \(a\);
- Zpg, a
list containing the coefficients in
zero-pole-gain form
- Sos, a
list of series second order sections (biquads)
Although the Arma form is default for compatibility
reasons, the use of Sos and the accompanying filtering
function sosfilt is generally preferred.
A second issue that may occur when using IIR filters, is instability.
This may be the result of numerical rounding errors or because too many
filter coefficients were used. Pole-Zero
Analysis can be useful here. A filter is stable if its impulse
response \(h(n)\) decays to 0 when
\(n\) increases. In terms of poles and
zeros, this is true if all of the filter’s poles are inside the unit
circle in the \(z\)-plane (Smith,
J.O. (2012)). The package gsignal offers the function
zplane that displays a filter’s poles and zeros in the
complex \(z\)-plane, as the following
figure illustrates. In the figure the ’0’s represent the zeros, and the
’X’s the poles.
op <- par(no.readonly = TRUE)
n <- layout(matrix(c(1, 2, 3, 3), nrow = 2, byrow = TRUE))
stable <- butter(3, 0.2, "low", output = "Zpg")
# artificially adapt pole
instable <- stable
instable$p[2] <- instable$p[2] - 2
zplane(stable, main = "Stable")
zplane(instable, main = "Instable")
t <- seq(0, 1, len = 100)
x <- sin(2* pi * t * 2.3) + 0.5 * rnorm(length(t))
z1 <- filter(stable, x)
z2 <- filter(instable, x)
plot(t, x, type = "l", xlab = "", ylab = "")
lines(t, z1, col = "green", lwd = 2)
lines(t, z2, col = "red")
legend("bottomleft", legend = c("Original", "Stable", "Instable"),
lty = 1, col = c("black", "green", "red"), ncol = 3)
Compensating for filter delay in IIR filters
Because the phase of the frequency response of IIR filters is not
linear, the filter delay cannot be easily compensated for as in the FIR
case. Recall that the 40-tap 30 Hz low-pass FIR filter used above for
filtering the ECG signal had a linear phase and a constant delay of 20
samples. If a 5th order elliptic low-pass filter at 30 Hz is used, it
can easily be seen that its phase is not linear (Figure (a) below), and
hence the filter delay is dependent of frequency (Figure (b) below).
Note that the group delay is defined to be the negative first derivative
of the filter’s phase response.
op <- par(mfrow = c(2, 1))
ell <- ellip(5, 0.1, 60, 30/(fs/2), "low")
ellf <- freqz(ell, fs = fs)
argh <- Arg(ellf$h)
argh[which(is.na(argh))] <- 0
phase <- unwrap(argh)
plot(ellf$w, phase, type = "l", xlab = "Frequency (Hz)", ylab = "Phase",
main = paste("30 Hz 5th order elliptical low-pass IIR filter\n",
"phase response is not linear"))
gd <- grpdelay(ell, fs = fs)
#> Warning in grpdelay.default(filt$b, filt$a, ...): setting group delay to 0 at
#> singularity
plot(gd, main = paste("group delay depends on frequency\n",
"mean:", round(mean(gd$gd), 1), "samples"))
This means that the filter delay cannot be compensated for in the
same way as for the FIR filter. An alternative is to use the function
filtfilt, which applies forward and backward filtering and
thus compensates for the delay, as shown in the figure below.
f <- filter(ell, ecg)
ff <- filtfilt(ell, ecg)
plot(time, ecg, type = "l", xlab = "Time", ylab = "", xlim = c(0,2))
title(ylab = expression(paste("Amplitude (", mu, "V)")), line = 2)
lines(time, f, col = "red", lwd = 2)
lines(time, ff, col = "blue", lwd = 2)
legend("topright", legend = c("Original", "filter()", "filtfilt()"),
lty = 1, lwd = c(1, 2, 2), col = c("black", "red", "blue"))
4. Filtering and Convolution
The most straightforward way to implement a digital filter is by convolving the
input signal with the filter’s impulse response. The package
gsignal contains several functions for convolution. The
function conv returns the 1-D convolution of two vectors
a and b in 3 ‘shapes’; “full”, for which the
output vector has a length equal to
length(a) + length(b) - 1; “valid”, which only returns the
central part of the convolution with an output length of
length(a); or “valid”, which returns only those parts of
the convolution that are computed without the zero-padded edges (output
length max(length(a) - length(b) + 1, 0)). For example:
u <- rep(1, 3)
v <- c(1, 1, 0, 0, 0, 1, 1)
conv(u, v, "full")
#> [1] 1 2 2 1 0 1 2 2 1
conv(u, v, "same")
#> [1] 1 0 1
conv(u, v, "valid")
#> NULL
conv(v, u, "valid")
#> [1] 2 1 0 1 2
Two-dimensional convolution of two matrices can be computed by the
conv2 function. In this case, the size of the output matrix
is nrow(A) + nrow(B) - 1 by
ncol(A) + ncol(B) - 1 for “full” convolution,
nrow(A) by ncol(A) for “same”, and
max(nrow(A) - nrow(B) + 1, 0) by
max(ncol(A) - ncol(B) + 1, 0) for “valid”. The function
conv2 is implemented in C++ for speed.
For long series convolution may be sped up by making use of the fact
that convolution in the time domain is equivalent to multiplication in
the frequency domain. Thus, the two series may be padded to the same
length, converted to the frequency domain by FFT, multiplied point-wise,
and transformed back to the time domain. The function cconv
uses this approach. However, if one series is much longer than the other
(as in typical filtering operations), zero-padding the shorter series to
the length of the longer series may not be the most efficient method. In
such cases, even faster methods like the overlap-add method used by the
function fftconv may be useful. That’s the theory at
least…
short <- runif(20L)
long <- runif(1000L)
# convolve two long series
ll <- microbenchmark::microbenchmark(conv(long, long),
cconv(long, long),
fftconv(long, long))
plot1 <- ggplot2::autoplot(ll)
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
# convolve a short and a long series
sl <- microbenchmark::microbenchmark(conv(short, long),
cconv(short, long),
fftconv(short, long))
plot2 <- ggplot2::autoplot(sl)
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
gridExtra::grid.arrange(plot1, plot2, nrow = 2, ncol = 1)
One-dimensional Filtering in gsignal is performed by the
function filter. It is a direct form II transposed
implementation in C++ of the standard linear time-invariant difference
equation \[\sum_{k=0}^{N} a(k+1) y(n-k) +
\sum_{k=0}^{M} b(k+1) x(n-k) = 0; 1 \le n \le length(x)\] If a
matrix is passed to filter, its columns are filtered.
Two-dimensional filtering can be done using filter2. The
function fftfilt uses the overlap-add method and FFT
convolution for speed (but your mileage may vary), and
filtfilt uses forward and backward filtering to avoid
filter delay, as used above. If numerical stability of filter
coefficients is an issue, filter design using series second order
sections and filtering with the function sosfilt may be
used.
Filtering long series in chunks
The functions filter and sosfilt can retain
the filter state in order to process long data series in chunks. The
following piece of code shows how this is done. A long series is split
into two parts, which are processed sequentially. In the first call to
filter, the final conditions zf are asked to be returned
after filtering the first part, which are then passed to the second call
to filter as the initial conditions zi for the second part
of the series. The two filtered parts can then be concatenated for form
the entire filtered series without discontinuities.
N <- 10000L
long <- runif(N)
part1 <- long[1:(N / 2)]
part2 <- long[((N / 2) + 1):N]
b <- c(2, 3)
a <- c(1, 0.2)
y <- filter(b, a, long)
y1 <- filter(b, a, part1, 'zf')
y2 <- filter(b, a, part2, y1$zf)
yy <- c(y1$y, y2$y)
all.equal(y, yy)
#> [1] TRUE
Using initial conditions to avoid filter startup effects
Initial conditions can also be used to set the initial state of the
filter so that the output starts at the same value as the first element
of the signal to be filtered. The initial conditions for the filter can
be computed using the functions filter_zi, or
filtic, as shown in the following example.
t <- seq(-1, 1, length.out = 201)
x <- (sin(2 * pi * 0.75 * t * (1 - t) + 2.1)
+ 0.1 * sin(2 * pi * 1.25 * t + 1)
+ 0.18 * cos(2 * pi * 3.85 * t))
h <- butter(3, 0.05)
zi <- filter_zi(h)
## alternatively, use:
## lab <- max(length(h$b), length(h$a)) - 1
## zi <- filtic(h, rep(1, lab), rep(1, lab))
z1 <- filter(h, x)
z2 <- filter(h, x, zi * x[1])
plot(t, x, type = "l", xlab ="", ylab = "")
lines(t, z1, col = "red")
lines(t, z2$y, col = "green")
legend("bottomright", legend = c("Original signal",
"Filtered without initial conditions",
"Filtered with initial conditions"),
lty = 1, col = c("black", "red", "green"))
5. Power spectrum analysis
The power
spectral density (PSD) of a time series describes the distribution
of power (variance) into frequency components composing that signal. The
Fourier
Transform is a nonparametric method of decomposing a signal into its
frequency spectrum. The functions fft and ifft
compute the Discrete Fourier Transform with a fast algorithm, the FFT. A
parametric alternative for autoregressive (AR) models is available
through the functions ar_psd, pburg, or
pyulear. The following figure shows how both FFT- and
AR-based methods can discover the periodicities of 5 and 12 Hz in a
noisy signal.
op <- par(mfrow = c(3, 1))
fs <- 200
nsecs <- 10
lx <- fs * nsecs
t <- seq(0, nsecs, length.out = lx)
# signal of 5 Hz + 12 Hz + noise
x <- (sin(2 * pi * 5 * t)
+ sin(2 * pi * 12 * t)
+ runif(lx))
plot(t, x, type = "l", xlab = "Time (s)", ylab = "", main = "Original signal")
pw <- pwelch(x, window = lx, fs = fs, detrend = "none")
plot(pw, xlim = c(0, 20), main = "PSD estimate using FFT")
py <- pyulear(x, 30, fs = fs)
plot(py, xlim = c(0, 20), main = "PSD estimate using Yule-Walker")
Welch’s method
Welch
(1967) proposed a method to estimate the power spectrum that reduces
the variance of the spectrum (at the expense of decreasing frequency
resolution - remember, there is no free lunch) by splitting the signal
into (usually) overlapping segments and windowing each segment, for
instance by a Hamming window. The periodogram is then computed for each
segment, and the squared magnitude is computed, which is then averaged
for all segments. The spectral density is the mean of the modified
periodograms, scaled so that area under the spectrum is the same as the
mean square of the data. In case of multivariate signals, cross-spectral
density, phase, and coherence are also returned. The input data can be
demeaned or detrended, overall or for each segment separately.
The following figure shows two signals, a sine and a cosine of 5 Hz
with noise added. Sines and cosines are the same waveforms, the only
difference being that a cosine leads the sine wave by an amount of
90\(^{\circ}\) (\(\pi / 2\) radians). Hence, a plot of the
PSD should show identical shapes for both signals, but a phase
difference at \(\pi / 2\) radians
should be visible at 5 Hz, which should also produce a high magnitude
squared coherence (near 1) at that frequency (coherence reflects
constant phase differences).
op <- par(mfrow = c(3, 1))
fs <- 200
nsecs <- 100
lx <- fs * nsecs
t <- seq(0, nsecs, length.out = lx)
# sine and cosine of signal of 5 Hz noise
x1 <- cos(2 * pi * 5 * t) + runif(lx)
x2 <- sin(2 * pi * 5 * t) + runif(lx)
x <- cbind(x1, x2)
pw <- pwelch(x, fs = fs)
plot(pw, plot.type = "spectrum", yscale = "dB", xlim = c(0, 50),
main = "A sine and a cosine of 5 Hz have the same PSD")
legend("topright", legend = c("Cosine", "Sine"), lty = 1:2, col = 1:2)
rect(3, -35, 7, -4, border = "red", lwd = 3)
plot(pw, plot.type = "phase", xlim = c(0, 50),
main = expression(bold(paste("but differ ", pi/2, " radians in phase at 5 Hz"))))
rect(3, -pi, 7, pi, border = "red", lwd = 3)
plot(pw, plot.type = "coherence", xlim = c(0, 50),
main = "leading to coherence ~ 1 at 5 Hz")
rect(3, 0, 7, 1, border = "red", lwd = 3)
Time-frequency analysis
If your signal’s frequency content changes over time, the power
spectrum is of limited use. You might then want to use some form of
time-frequency analysis. Specialized R packages exist for wavelet
analysis capable of doing time-frequency analysis, such as wavelets, Rwave, and waveslim.
gsignal does contain a basic function for computing the
discrete wavelet transform (dwt), but is not otherwise
specialized for wavelet analysis.
Another way of decomposing a signal both in time and frequency is the
Short-Term
Fourier Transform, which can be calculated by the function
stft. Here, the data to be transformed is broken up into
(overlapping) chunks. Each chunk is then Fourier transformed, and added
to a record of magnitude and phase for each point in time and frequency.
Alternatively, the spectrogram (specgram) in essence does
the same - the spectogram is the squared magnitude of the STFT of the
signal.
The following figure represents the spectrogram of a
chirp signal, which is a signal in which the frequency
changes with time. By default, the specgram and
stft function produce grayscale plots, but here it is shown
how other color palettes can be used.
op <- par(mfrow = c(2, 1))
jet <- grDevices::colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F",
"yellow", "#FF7F00", "red", "#7F0000"))
sp <- specgram(chirp(seq(0, 5, by = 1/8000), 200, 2, 500, "logarithmic"), fs = 8000)
plot(sp, col = jet(20))
c2w <- grDevices::colorRampPalette(colors = c("red", "white", "blue"))
plot(sp, col = c2w(50))
