3 LMD
A multi-component signal can be broken down into its constituent PFs and a residue. Each PF is a mono component that is the result of an envelope signal and a frequency modulated signal. In LMD a two-tier method is utilized to perform the demodulation .First, the inner cycle is used to conduct a thorough computation of the PF. Second, in the outer cycle, the signal is decomposed depending on the iterations. In the following sections the methods have been broken down into steps that LMD technique requires to complete the analysis on a given signal x(t). Consider the following synthetic signal \(x (t)\) of the form :
\[x ( t ) = 0.25t + sin ( πt ) + sin ( 2πt ) + sin ( 6πt )\]
The above signal is represented in graphical form below:
fs <- 2000 #Sampling Frequency
time_component <- seq(from = 0, to = 9, by = 1 / fs)
# Simulated Signal
signal <- (0.25 * time_component + sin(pi * time_component) + sin(2 * pi * time_component) +
sin(6 * pi * time_component))
dummy_df <- data.frame(Time = time_component, Value = signal)
p1 <- ggplot(dummy_df) +
aes(x = Time, y = Value) +
geom_spline(size = 2, colour = "#1D2B71") +
scale_color_hue() +
labs(title = "Fig 1. Sample Signal") +
theme_minimal() +
theme(legend.position = "none")
plot(p1)
3.1 Steps for PF Extraction
3.1.1 Step 1
In the first step, all the local extrema \(n_i\) are extracted from the original
signal \(x(t)\). The extreme values of
a signal can be found out by using the find_extrema
function from LMD.
LMD::find_extrema(signal)find_extrema function expects following parameters:
signal: Signal is a vector of numeric valuesinclude_endpoints: A logical value indicating whether to include end points or not. Default value for this is set to TRUE.
The function returns indexes of all extreme values. In the below chunk the extreme values of the original sample signal x(t) have been indicated by red color.
extreme_values <- LMD::find_extrema(signal)
p2 <- ggplot() +
geom_spline(
data = dummy_df,
aes(x = Time, y = Value),
size = 2,
colour = "#1D2B71"
) +
geom_point(aes(x = dummy_df$Time[extreme_values], y = dummy_df$Value[extreme_values]),
size = 5,
colour = "red") +
scale_color_hue() +
labs(title = "Fig 2. Sifting Process: (1)") +
theme_minimal() +
theme(legend.position = "none")
plot(p2)
3.1.2 Step 2
In the second step the mean value \(m_i\) is calculated by taking average of the current extrema \(n_i\) and the next extrema \(n_{i+1}\). So the \(i_{th}\) mean value \(m_i\) of each two successive extrema \(n_i\) and \(n_{i+1}\) is given by:
\[ \begin{aligned} &m_i = {\frac{n_i+n_{i+1}}{2}} \\ \end{aligned} \]
All the mean values \(m_i\) of two successive extreme are connected by straight lines, as shown below:
extreme_values <- LMD::find_extrema(signal)
ma_enve = local_mean_and_envelope(signal, extreme_values)
mean = ma_enve$mean
p3 <- ggplot() +
geom_spline(
data = dummy_df,
aes(x = Time, y = Value),
size = 2,
colour = "#1D2B71",
alpha = 0.5
) +
geom_point(
aes(x = dummy_df$Time[extreme_values], y = dummy_df$Value[extreme_values]),
size = 5,
colour = "red",
alpha = 0.5
) +
geom_line(aes(x = dummy_df$Time, y = mean),
size = 1,
colour = "red") +
scale_color_hue() +
labs(title = "Fig 3. Sifting Process: (2.1)") +
theme_minimal() +
theme(legend.position = "none")
plot(p3)
The local mean function \(m_{11}(t)\) is derived by taking a moving weighted average of the means \(m_i\).
ma = ma_enve$ma
p4 <- ggplot() +
geom_spline(
data = dummy_df,
aes(x = Time, y = Value),
size = 2,
colour = "#1D2B71",
alpha = 0.5
) +
geom_point(
aes(x = dummy_df$Time[extreme_values], y = dummy_df$Value[extreme_values]),
size = 5,
colour = "red",
alpha = 0.5
) +
geom_line(aes(x = dummy_df$Time, y = ma),
size = 1,
colour = "red") +
scale_color_hue() +
labs(title = "Fig 4. Sifting Process: (2.2) ") +
theme_minimal() +
theme(legend.position = "none")
plot(p4)
Similarly the envelope value\(a_i\) is given by:
\[ \begin{aligned} &a_i = {\frac{|n_i-n_{i+1}|}{2}} \\ \end{aligned} \]
envelope = ma_enve$enve
Time=dummy_df$Time
p5 <- ggplot() +
geom_line(aes(x = Time, y = envelope),
size = 1,
colour = "red") +
scale_color_hue() +
labs(title = "Fig 5. Sifting Process: (2.3)") +
theme_minimal() +
theme(legend.position = "none")
plot(p5)
The envelope values \(a_i\) are smoothed using weighted moving average and the envelope function \(a_{11}\)(t) is formed.
enve_sm = ma_enve$enve_sm
Time=dummy_df$Time
p6 <- ggplot() +
geom_line(aes(x = Time, y = envelope),
size = 0.5,
colour = "red",alpha=0.5) +
geom_line(aes(x = Time, y = enve_sm),
size = 1,
colour = "red") +
scale_color_hue() +
labs(title = "Fig 6. Sifting Process: (2.4) ") +
theme_minimal() +
theme(legend.position = "none")
plot(p6)
The local mean and envelope and corresponding smoothed values can be
found out by using the local_mean_and_envelope function
from LMD.
LMD::local_mean_and_envelope(signal, extrema)local_mean_and_envelope function expects following
parameters:
signal: Signal is a vector of numeric valuesextrema: A vector of numeric values for indexes for extreme values in the signal
The function returns local mean, moving averaged local mean, envelope and moving averaged local envelope values.
3.1.3 Step 3
In the step 3 we calculate the zero-mean signal \(h_{11}\)(t) by taking a difference of the original signal \(x(t)\) and the local mean function \(m_{11}\)(t) :
\[ \begin{aligned} &h_{11}(t) = {x(t)-m_{11}(t)} \\ \end{aligned} \]
3.1.4 Step 4
Then, a frequency modulated signal \(s_{11}\)(t) is calculated as:
\[ \begin{aligned} &s_{11}(t) = {h_{11}(t)/a_{11}(t)} \\ \end{aligned} \]
3.1.5 Step 5
Repeat the above steps 1 to 3 to get the envelope value \(a_{12}\)(t) of \(s_{11}\)(t) . This is the checkpoint for the inner loop, if this envelope function \(a_{12}\)(t) is equal to 1, the inner iteration is stopped and \(s_{12}\)(t) is considered as the first purely frequency modulated (FM) signal. Else, the \(s_{11}\)(t) is considered as the original signal and Steps 1 to 4 are repeated n times until the condition \(a_{1(n+1)}\)(t)=1 is satisfied for signal \(s_{1(n)}\)(t).
3.1.6 Step 6
Calculate the envelope signal \(a_1\)(t) of the first PF1 , multiply all the smoothed local envelopes from the iterations:
\[ \begin{aligned} & a_1(t) = a_{11}(t) * a_{12}(t)*a_{13}(t)\dotso a_{1n}(t) \\ \end{aligned} \]
3.1.7 Step 7
Compute the first \(PF_1\) using \(a_{1}(t)\) and \(s_{1n}(t)\):
\[\begin{aligned} & PF_1 = a_{1}(t)s_{1n}(t) \\ \end{aligned} \]
Now the residual signal is given by \(u_1(t)\) = \(x(t)\)-\(PF_1(t)\) . The residual \(u_1(t)\) is considered as the original signal and the above steps are repeated n times until \(u_n(t)\) does not contain oscillation.
This is the called outer loop of LMD. Finally, the signal x(n) is represented as a sum of PFs and residual signal:
\[\begin{aligned} & x(n) = \sum^q_{i=1}PF_i(n)+u_q(n) \\ \end{aligned} \]
The LMD method can be applied on a signal by using the
lmd function from LMD Package.
LMD::lmd(signal)lmd function expects following parameters:
signal: Signal is a vector of numeric valuesinclude_endpoints: A boolean value indicating whether to treat the endpoint of the signal as a pseudo-extreme pointmax_smooth_iteration: An integer value for maximum number of iterations of moving average algorithmmax_envelope_iteration: An integer value for maximum number of iterations when separating local envelope signalsenvelope_epsilon: A float value to terminate processing when obtaining pure FM signalconvergence_epsilon: A float value to terminate processing when modulation signal convergesmax_num_pf: An integer value for maximum number of PFs to be generated
The function returns PF and residual value as a list.
3.2 Flowchart of LMD
A flowchart of the LMD method is presented in figure below for better
understanding.
Figure 7: Flowchart of LMD
