Time Frequency Analysis for Radio Frequency (RF) Signal Processing

Abstract

In high-density radio frequency (RF) signal environments, receivers usually acquire signals from multiple sources. These RF signals may be co-channel and co-duration, which cause significant difficulties for processing them. Time-Frequency analysis combined with a projection pursuits graph approach provides an effective way to detect, separate, and classify these multiple source RF signals. Time-frequency analysis includes a spectrogram approach and a scalogram approach. The feature points on the instantaneous frequency function of a frequency modulation radio frequency (FMRF) signal can be extracted from either the spectrogram or scalogram of this FMRF signal. With the projection pursuits graph approach, these feature points are grouped into time-frequency functions to represent the multiple components for the separation, detection, and classification of this multisource FMRF signal.

1. Introduction

In congested electromagnetic environments, the radio frequency (RF) signals a receiver receives may include multiple time-frequency overlapped signals transmitted from multiple emitters. Traditional RF signal processing techniques may have difficulty separating and processing these multisource signals.

The instantaneous frequency function of an FMRF signal can be approximately modeled by low-order polynomials. Chirplet transforms and polynomial chirplet transforms have been investigated to process multisource FMRF signals [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These approaches separate and process multisource cochannel FMRF signals effectively; however, their implementations are expensive due to high dimensional transforms.

With a first-order polynomial approximation of the phase function of an FMRF signal, the short-time Fourier transform approach provides a simple and low-cost implementation for instantaneous frequency estimation. Unlike chirplet and polynomial chirplet transforms which need to perform transforms from time to high dimensional frequency and chirp spaces, the short-time Fourier transform approach creates spectrograms and only needs to perform time to frequency transforms. Using fast Fourier transforms, the short-time Fourier transform for a local window with size W only needs $O\left(\mathit{\text{Wlog}}W\right)$computations, which are much lower than utilizing chirplet or polynomial chirplet transform approaches [14, 15].

Spectrograms are created by a fixed window size Fourier transform. For a low-frequency component, it needs a large window to capture enough changes for this low-frequency component. However, for a high-frequency component, it needs a small window to have a high time resolution. The constant window size for spectrogram cannot satisfy these conflict requirements. To address this issue, a natural extension is to perform Fourier transforms with changeable window sizes. For high-frequency components, small window sizes are used to perform transforms while large window sizes are used for low-frequency transforms. This extension leads to the wavelet transforming with constant weights in the window, creating a scalogram. The weight functions could also be other functions that lead to different wavelet transforms. For instance, choosing a Gaussian function creates Gabor or Morlet wavelet transform.

Spectrograms or scalograms provide the time-frequency representation of a multisource FMRF signal. Separating this multisource FMRF signal into each independent source component needs further processing. The ridge points of spectrograms or scalograms over some thresholds generate the points for instantaneous frequency functions. In this chapter, a connected graph will be introduced to extract instantaneous frequency functions when they are not crossed with each other. When the instantaneous frequency functions are crossed with each other, a projection-pursuit approach is described to separate and extract these instantaneous frequency functions.

2. A FMRF signal model and its spectrogram

In this section, an FMRF signal model with a single source is introduced, and a $\text{Sinc}\left(\right)$function for its spectrogram is derived from this model with short-time Fourier transforms. Then, this FMRF signal model and its spectrogram computation are extended to multiple component FMRF signals.

2.1 An FMRF signal model with a single component

A single component FMRF signal is described by the following model,

$$s\left(t\right)=A_{0}exp\left(\mathit{i\phi }\left(t\right)\right)+n\left(t\right)$$E1

where $\phi \left(t\right)$ is the instantaneous phase function of this FMRF signal and $n\left(t\right)$ represents additive white noises.

Another function to describe an RF signal is its instantaneous frequency function $f\left(t\right)$. The frequency function $f\left(t\right)$ and phase function $\phi \left(t\right)$ have differential and integral relations are shown as follows,

$$f\left(t\right)=\frac{d}{\mathit{dt}}\phi \left(t\right)\text{and}\phi \left(t\right)={\phi }_0+\int f\left(t\right)\mathit{dt}$$E2

It is shown in Eq. (2) that the frequency function $f\left(t\right)$ of an RF signal represents its phase function $\phi \left(t\right)$ with only a constant phase uncertainty ${\phi }_0$. Due to this reason, in this chapter, we will focus on applying the frequency functions to analyze and process FMRF signals.

The instantaneous frequency function $f\left(t\right)$ is a one-dimensional manifold imbedded into a two-dimensional time-frequency space. Without noises ($n\left(t\right)=0$), a single component FMRF signal and its time-frequency manifold are shown in Figure 1, where the left side is an FMRF signal, and the right side is its time-frequency manifold.

Figure 1.

An FMRF signal (left) and its time-frequency manifold (right).

Since the time-frequency manifold of an FMRF signal is the representation of this FMRF signal, we can use time-frequency manifolds to classify or recognize RF signals. Also, the time-frequency manifolds of an FMRF signal provide an estimation of its instantaneous frequencies.

2.2 Sinc() function of the time-frequency image or spectrogram of a single component FMRF signal

The time-frequency image or spectrogram of an FMRF signal $s\left(t\right)$ is the magnitude of the short-time Fourier transform of $s\left(t\right)$,

$$I\left(\omega ,t\right)=\left|\sum _{\tau =0}^{W-1}{A}_{0}exp\left(\mathit{i\phi }\left(t+\tau \right)\right)e^{-\mathit{j\omega \tau }}+n_I(\omega ,t)\right|$$E3

where W is the window size of this short-time Fourier transform.

Expanding $\phi \left(t+\tau \right)$ by its first-order Taylor series around t in a local window, we have

$$\phi \left(t+\tau \right)=\phi \left(t\right)+c_1\left(t\right)\tau +O\left({\tau }^2\right)$$E4

where $c_1\left(t\right)=\frac{\mathit{d\phi }\left(t\right)}{\mathit{dt}}$, which is the instantaneous frequency of $s\left(t\right)$ at time t, and the first-order Taylor expansion in (4) is a linear approximation to the phase function $\phi \left(t\right)$ in its local window.

Under the linear approximation of a phase function, the time-frequency image or spectrogram of $s\left(t\right)$ can be derived from (3) and (4),

$$\begin{array}{rcl}I\left(\omega ,t\right)& \approx & \left|\sum _{\tau =0}^{W-1}{A}_{0}exp\left(\mathit{i\phi }\left(t\right)\right)e^{j\left(c_1\left(t\right)-\omega \right)\tau }+n_I\left(\omega ,t\right)\right|\\ & =& \left|A_{0}exp\left(\mathit{i\phi }\left(t\right)\right)\right|\left|\sum _{\tau =0}^{W-1}{e}^{j\left(c_1\left(t\right)-\omega \right)\tau }+n_I\left(\omega ,t\right)\right|=\left|\frac{A_{0}sin\left(\frac{W\left(c_1\left(t\right)-\omega \right)}{2}\right)}{sin\left[\left(c_1\left(t\right)-\omega \right)/2\right]}+n_I\left(\omega ,t\right)\right|\end{array}$$

Thus, we have approximated the spectrogram of an FMRF signal

$$I\left(\omega ,t\right)\approx \mid \frac{A_{0}sin\left(W\left(c_1\left(t\right)-\omega \right)/2\right)}{sin\left[\left(c_1\left(t\right)-\omega \right)/2\right]}+n_I\left(\omega ,t\right)\mid$$E5

Eq. (5) shows that when noises $n_I\left(\omega ,t\right)$ are low, at a given time t, the spectrogram of an FMRF signal is a Sinc() function in the frequency direction. This Sinc() function reaches its maximum at the instantaneous frequency $c_1\left(t\right)$.

The spectrogram of the FMRF signal in Figure 1 is shown in Figure 2. Figure 2 shows the Sinc() patterns in the vertical (frequency) direction.

Figure 2.

An FMRF signal and its time-frequency image (spectrogram).

2.3 Spectrograms of multisource and cochannel multisource and cochannel FMRF signals

The multisource and co-channel FMRF signals received by a receiver is modeled as

$$x\left(t\right)=n\left(t\right)+\sum _{k=1}^K{s}_k\left(t\right)=n\left(t\right)+\sum _{k=1}^K{a}_{k}exp\left({\mathit{i\phi }}_k\left(t\right)\right)$$E6

where, K is the number of sources for the FMRF signal, and n(t) is the noises of the receiver.

Similar to (4), a linear approximation in a local window is used to approximate the phases for the multisource FMRF signal,

$${\phi }_k\left(t+\tau \right)={\phi }_k\left(t\right)+c_{1,k}\left(t\right)\tau +O\left({\tau }^2\right)$$E7

where $c_{1,k}\left(t\right)$ is the instantaneous frequency for the component signal $s_k\left(t\right)$.

An equation to compute the spectrogram for the multisource FMRF signal is derived by substituting (7) into (6),

$$I\left(\omega ,t\right)=\mid \sum _{k=1}^K\frac{A_{k}sin\left(\frac{W\left(c_{1,k}\left(t\right)-\omega \right)}{2}\right)}{sin\left[\left(c_{1,k}\left(t\right)-\omega \right)/2\right]}+n_I\left(\omega ,t\right)\mid$$E8

where, $n_I\left(\omega ,t\right)$ is the spectrogram noises. (8) shows that when noises $n_I\left(\omega ,t\right)$ are low, the spectrogram of this multisource FMRF signal is the magnitude of the summation of multiple Sinc() functions.

A multisource and cochannel FMRF signal is shown in Figure 3. The right side of Figure 3 shows the spectrogram of this FMRF signal where the Sinc() function patterns are distributed in the frequency direction.

Figure 3.

A multisource and cochannel FMRF signal and its spectrogram. The left is the FMRF signal, the middle is its instantaneous frequency function, and the right is its spectrogram.

3. A scalogram as an extension of spectrogram

It is shown in Section 2 that the spectrogram of an FMRF signal created by short-time Fourier transform (STFT) demonstrates Sinc() patterns. In this section, we will show that the wavelet transforms with a uniform window is a direct extension of the STFT, so the scalogram generated by wavelet transforms is a direct extension of the spectrogram.

3.1 Scalogram computaion of a single component FMRF signal

Define a rectangle window function

$$R\left(t\right)=\left\{\begin{array}{c}1,0\le t<1\\ 0,\mathit{\text{otherwise}}\end{array}\right.$$E9

For a window size W, we have

$$R\left(\frac{t}{W}\right)=\left\{\begin{array}{c}1,0\le t<W\\ 0,\mathit{\text{otherwise}}\end{array}\right.$$E10

With the help of the window function $R\left(\frac{t}{W}\right)$, the STFT in (3) is written into the following format,

$$I\left(\omega ,t\right)=\left|\sum _{\tau =-\infty }^{\infty }{A}_{0}exp\left(\mathit{i\phi }\left(t+\tau \right)\right)R\left(\frac{\tau }{W}\right)e^{-\mathit{j\omega \tau }}+n_I(\omega ,t)\right|$$E11

The computation of spectrograms in (11) is the same as that in (3). They both give the same STFT for spectrogram computations by a uniform distributed weight function $R\left(\frac{t}{W}\right)$ with a fixed window size W.

When the window size W in (11) is chosen to be changed by $W=\frac{c}{\omega }$ as frequency $\omega$ changes,

$$I\left(\omega ,t\right)=\left|\sum _{\tau =-\infty }^{\infty }{A}_{0}exp\left(\mathit{i\phi }\left(t+\tau \right)\right)R\left(\frac{\tau }{\frac{c}{\omega }}\right)e^{-\mathit{j\omega \tau }}+n_I(\omega ,t)\right|$$E12

where $c$ is a constant.

Eq. (12) is a wavelet transform with a mother wavelet $R\left(\frac{\tau }{c}\right)e^{-\mathit{j\tau }}$, and $\frac{1}{\omega }$ is the scale of the wavelet transform. $A$ uniform distributed weight function is discussed in this chapter. $R\left(\right)$ can be a different weight function. If $R\left(\right)$ is chosen to be a Gaussian function, (12) will give Gabor or Morlet wavelet transform.

To distinguish scalogram from spectrogram, we change $I\left(\omega ,t\right)$ to $W\left(\omega ,t\right)$ to show that the scalogram is generated by a wavelet transform. The scalogram of an FMRF signal is calculated by the following equation.

$$W\left(\omega ,t\right)=\left|\sum _{\tau =-\infty }^{\infty }{A}_{0}exp\left(\mathit{i\phi }\left(t+\tau \right)\right)R\left(\frac{\mathit{\omega \tau }}{c}\right)e^{-\mathit{j\omega \tau }}+n_W(a,t)\right|$$E13

(11) and (13) show the close relationship between STFT and the wavelet transform. The scale in the wavelet transform is inversely proportional to the frequency while the scale STFT is fixed. In other words, the wavelet transform can be treated as an adaptive STFT where the window size of the STFT (referred to as scale in the wavelet transform) adapts to the frequency change of the STFT. When the frequency is high, the window size is small so as to catch the high resolution in time. When the frequency is low, the window size is large so as to obtain a high resolution in frequency. In this sense, a wavelet transform usually creates a higher performance than an STFT due to the wavelet’s adaptive properties.

Similar to the derivation of the spectrogram calculation by summation in (3), the scalogram calculation can also be derived using wavelet transforms. Writing (13) into a summation format creates the following expression,

$$W\left(\omega ,t\right)=\left|\sum _{\tau =0}^{\frac{\alpha }{\omega }-1}{A}_{0}exp\left(\mathit{i\phi }\left(t+\tau \right)\right)e^{-\mathit{j\omega \tau }}+n_I(\omega ,t)\right|$$E14

The scalogram calculated by (13) is further simplified by substituting the FMRF signal of (4) into (14),

$$W\left(\omega ,t\right)\approx \left|\sum _{\tau =0}^{\frac{\alpha }{\omega }-1}{A}_0{e}^{\mathit{j\phi }\left(t\right)}{e}^{j\left(c_1\left(t\right)-\omega \right)\tau }+n_I\left(\omega ,t\right)\right|$$E15

The $e^{\mathit{j\phi }\left(t\right)}$ in (15) does not depend on $\tau$. Thus, (15) can be written into another form,

$$W\left(\omega ,t\right)=\mid e^{\mathit{j\phi }\left(t\right)}\mid \left|A_0\sum _{\tau =0}^{\frac{\alpha }{\omega }-1}{e}^{j\left(c_1\left(t\right)-\omega \right)\tau }+e^{-\mathit{j\phi }\left(t\right)}{n}_I\left(\omega ,t\right)\right|=\left|A_0\sum _{\tau =0}^{\frac{\alpha }{\omega }-1}{e}^{j\left(c_1\left(t\right)-\omega \right)\tau }+e^{-\mathit{j\phi }\left(t\right)}{n}_I\left(\omega ,t\right)\right|$$E16

If noise term $e^{-\mathit{j\phi }\left(t\right)}{n}_I\left(\omega ,t\right)$ is denoted as $n_W\left(\omega ,t\right)$, the wavelet transform in (16) becomes,

$$W\left(\omega ,t\right)=\mid \frac{A_{0}sin\left(\frac{\alpha }{\omega }\frac{c_1\left(t\right)-\omega }{2}\right)}{sin\left(\frac{c_1\left(t\right)-\omega }{2}\right)}+n_W\left(\omega ,t\right)\mid$$E17

Eq. (17) shows that similar to the spectrogram $I\left(\omega ,t\right)$, the scalogram $W\left(\omega ,t\right)$ also demonstrates the Sinc() properties near the instantaneous frequency $c_1\left(t\right)$. The difference is that the Sinc() function of the spectrogram $I\left(\omega ,t\right)$ oscillates in an equal period while the scalogram $W\left(\omega ,t\right)$ oscillates in an increasing period as frequency increases.

The comparison between spectrogram and scalogram is shown in Figure 4. In Figure 4, the frequency of the FMRF signal is chosen as 10 kHz in the local window. For the spectrogram, the window size is chosen as 20. For the scalogram, the window size is selected to change from 18 to 22. At the center frequency 10 kHz, the mask size of the scalogram is the same as the window size for spectrogram 20. Figure 4 shows that the Sinc() function oscillates with the same frequency in the frequency direction for spectrogram. However, the oscillation frequency for the scalogram increases from low frequency to high frequency.

Figure 4.

The Sinc() function for spectrogram and Scalogram.

3.2 Scalogram computaion of a multisource FMRF signal

Similar to the computation of a single source FMRF signal, the scalogram computation of a multisource FMRF signal is given by replacing the fixed-size window summation in (8) with the frequency-dependent window summation,

$$W\left(\omega ,t\right)=\mid \sum _{k=1}^K\frac{A_{k}sin\left(\frac{\alpha }{\omega }\frac{\left(c_{1,k}\left(t\right)-\omega \right)}{2}\right)}{sin\left(\frac{\left(c_{1,k}\left(t\right)-\omega \right)}{2}\right)}+n_W\left(\omega ,t\right)\mid$$E18

Eq. (18) shows that the scalogram of each component of a multisource FMRF signal is a Sinc() function in the local frequency direction, which is similar to the spectrogram.

4. Connected graph approach for spectrogram and scalogram

Both spectrogram and scalogram are two-dimensional images and both have similar Sinc() function properties. Their image processing techniques are also similar. Therefore, only the spectrogram processing technique is introduced in this chapter to demonstrate the connected graph approach for FMRF signal processing. The scalogram processing is a straightforward extension.

4.1 Sparse cloud point representation of spectrograms

By binarizing a spectrogram, we can create a sparse cloud point representation (a binary image) of this spectrogram and call it the sparse time-frequency map. Thresholding and local maximum in the frequency direction can be used to create this sparse time-frequency map

$$M\left(\omega ,t\right)=\left\{\begin{array}{cc}I\left(\omega ,t\right),& \mathit{\text{if}}I\left(\omega ,t\right)>I\left(\omega -1,t\right),I\left(\omega -1,t\right)\mathit{\text{and}}T\\ 0,& \mathit{\text{otherwise}}\end{array}\right.$$E19

An FMRF signal, its spectrogram, and its sparse time-frequency map is shown in Figure 5. Figure 5 shows that the nonzero points in the sparse time-frequency map created from the spectrogram of an FMRF signal form the time-frequency manifold that represents this FMRF signal. Since the nonzero pixels are a very small portion of the entire image of pixels and the connected graph approach, we are using only performs on these nonzero pixels, this connected graph approach has a very low computational cost.

Figure 5.

An FMRF signal, and its spectrogram and sparse time-frequency map.

4.2 The spectrogram and sparse time frequency map of a Noisy FMRF signal

We have discussed the spectrogram and sparse time-frequency map with no noises as shown in Figure 5. The spectrogram and its sparse time-frequency map for a noisy FMRF signal is shown in Figure 6.

Figure 6.

Spectrogram and sparse time-frequency map of an FMRF signal with different noise levels: the top is for signal to noise ratio (SNR) = 6DB, and the bottom for SNR = 0 DB.

Figure 6 shows that the spectrograms and time-frequency maps for very noisy FMRF signals are similar to those without noises in Figure 5. The difference is that the time-frequency maps for noisy signals add some extra noise pixels. These noise pixels will be removed by the connected graph approach, however.

4.3 A graph approach for extracting time frequency manifolds

Figures 5 and 6 show that the sparse time-frequency map of an RF signal includes the points on the time-frequency manifold of this RF signal. A connected graph approach is used to extract this time-frequency manifold.

The graph to represent the sparse time-frequency map consists of nodes and edges. Each node $n_i$ of the graph, corresponding to a nonzero pixel of the sparse time-frequency map, is represented by three variables. The first two variables x and y represent the pixel position for this node. The third member is a list of its neighbor nodes that are used to build connected graphs. A node $n_i$ is defined with C++ as

Two nodes are connected if they are neighbors. For the node $n_i$, its neighbor nodes are found by checking the distance between two nodes. $n_j$ is the neighbor of $n_i$ if $\left(n_i.x-n_j.x\right)\ast \left(n_i.x-n_j.x\right)+\left(n_i.y-n_j.y\right)\ast \left(n_i.y-n_j.y\right)$< Threshold Distance.

Each node is connected to its neighbors but disconnected to non-neighbor points. With this graph, the connected components can be found. Obviously, some connected graphs are the time-frequency manifolds as the FMRF signal, while others could be noises. Usually, small connected graphs are noises and can be removed.

The time-frequency manifolds for a two-source FMRF signal are extracted and shown in Figure 7, where two connected graphs are displayed for the time-frequency manifolds (red and blue) for two FMRF signals components.

Figure 7.

A two-source mixed FMRF signal and its time-frequency manifolds. These time-frequency manifolds are extracted by the connected graph approach from the sparse time-frequency map shown in the bottom right of Figure 5.

Figure 7 shows that each individual component (red and blue) of the two-source cochannel and co-duration FMRF signals can be extracted using the connected graph approach.

4.4 Issues for the graph techniques

If the two components in a two-source FMRF signal are not connected to each other in their sparse time-frequency map, the connected graph approach is capable of extracting, separating, and classifying them, as is shown in Figure 7. However, when two or multiple components are connected to each other, as shown in Figure 8, the connected graph approach may not work well.

Figure 8.

A time-frequency manifold connected two source co-channel and co-duration FMRF signal, its time-frequency manifolds, spectrogram, and extracted manifolds.

Figure 8 shows a two-source FMRF signal, its time-frequency manifold, spectrogram, and the time-frequency manifolds extracted by the graph approach. One of these two source signals is a linear frequency modulation signal with a negative sweep rate (frequency decrease), and the other one is a nonlinear frequency modulation signal with a positive sweep rate (frequency increase). These two source FMRF signals are overlapped in both time and spectral space and form pulse-in-pulse signals. As is demonstrated in Figure 8, the connected graph approach cannot separate these two connected time-frequency manifolds. This inseparable problem causes serious issues for classifications and other RF signal processing. In the next section, a projection pursuit approach will be discussed to address this issue.

5. Projection pursuits approach for pulse-on-pulse FMRF signal processing

When the time-frequency manifolds of two FMRF components are crossed with each other, the spatial distance-based neighbor point definition has problems. These problems and their possible solutions are shown in Figure 9.

Figure 9.

Spatial neighbors and string neighbors. Spatial neighbors are defined by spatial distances and string neighbors are defined by both spatial distances and strings.

In Figure 9, the left figure defines the neighbor points in the graph approach by spatial distances. We call these neighbor points the spatial distance neighbor points. In this definition, the two manifolds are inseparable. Different from the spatial neighbor approach, a string neighbor point approach is used to build time-frequency manifolds. Two points are neighbors if these two points are spatial neighbors and if they are on the same string. The string neighbor approach is shown on the right side of Figure 9. Figure 9 shows that the two manifolds are separable with the string neighbor approach even though they are inseparable from the spatial neighbor approach.

The projection pursuit approach is used to create string neighbor points. This approach is implemented in the following two steps:

Step 1. Create a graph for the time-frequency map by the spatial distance approach.

For each nonzero pixel, create a node $n_i$ . For M nonzero pixels,$i=0\sim \left(M-1\right)$. The x, y member of a node $n_i$ is the location of the pixel. The neighbor point set $N_i$ of this node are generated by the spatial distance approach.

Step 2. Refine the neighbor points of each node $n_i$ with the string approach.

At the location of each node $n_i$, draw K lines $l_1,l_2,\dots \dots l_K$ with an increment angle 0, $\frac{180}{K},2\ast \frac{180}{K},\dots \dots \left(K-1\right)\ast \frac{180}{K}$ to the horizon axis. Compute the projection to $l_1,l_2,\dots \dots l_K$ for each node of the neighbor nodes $N_i$. For each $l_i\left(i=1\sim K\right),$add the top M (< number of nodes in $N_i)$ highest projections as the score $S_i$. Choose $l_m$ so that its score $S_m$ is largest. Then remove the neighbor nodes that have low projection values to $l_m$. After all the removing operations, the remaining neighbor nodes are the string neighbor nodes.

The above two steps are used to create string neighbor nodes. After the string neighbor nodes of the graph are created, the same connected graph approach discussed in Section 4 is used to create connected graphs and build the time-frequency manifolds for the FMRF signals.

The test results for the projection pursuits approach are shown in Figure 10. The right side of Figure 10 shows two-time-frequency manifolds extracted by the projection pursuits approach. The red line is the down sweep linear frequency modulation component of this two-source FMRF signal while the white curve is the time-frequency manifold of the down sweep nonlinear frequency modulation component. It is shown from these test results that the projection pursuit approach is capable to separate and extract the time-frequency manifolds of complicated multisource FMRF signals (Figure 10).

Figure 10.

A two-source mixed FMRF signal and its time-frequency manifolds extracted by graph and projection pursuits.

6. Computational complexity analysis

Both spectrogram and scalogram approaches involve three components to perform their FMRF signal processing: transformation from an FMRF signal to a two-dimensional image, binarization of the image, and graph projection pursuit for creating the manifold of the FMRF signal.

Assume that the length of the signal to process is N. For the spectrogram approach, the transform from the FMRF signal to the spectrogram takes 0(NlogW) operations for a wind size W. The image size is W*(N/W) = N. Thus, the binarization takes 0(N) operations. Since the projection pursuits approach only processes a small fractional number of points in the image, its computational cost is much lower than 0(N). Putting the implementation of these three components together leads to the computational complexity 0(NlogW) for the spectrogram approach. Thus, the computation cost for the binarization and graph pursuits approach could be ignored when compared to the transform to create the spectrogram.

For the scalogram approach, since the transform from the FMRF signal to its scalogram image has a higher computational cost than the spectrogram approach and the same methods as the spectrogram approach are used for the binarization and graph projection pursuit, the computational complexity for the scalogram approach is the same as the computational complexity of the scalogram generation from the FMRF signal.

7. Conclusion

In this chapter, we introduce the spectrogram generation of an FMRF signal by using short-time Fourier transforms. Then, the spectrogram computation approach is extended to the scalogram computation by replacing the fixed size masks with frequency dependent masks.

Both spectrograms and scalograms are images, and a projection pursuits approach is introduced to process these images for separating and processing multisource cochannel and co-site FMRF signals.

It is shown that the projection pursuits method is very efficient, and its computational cost can be ignored when compared to the spectrogram or scalogram generation. Also, the projection pursuits approach is robust. It can separate and extract both non-connected and connected time-frequency manifolds for FMRF signal processing.