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Discrete Wavelet Transform

A Signal Processing Approach
D. Sundararajan(Autor*in)
Wiley (Verlag)
1. Auflage
Erschienen am 3. August 2015
344 Seiten
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Provides easy learning and understanding of DWT from a signal processing point of view
* Presents DWT from a digital signal processing point of view, in contrast to the usual mathematical approach, making it highly accessible
* Offers a comprehensive coverage of related topics, including convolution and correlation, Fourier transform, FIR filter, orthogonal and biorthogonal filters
* Organized systematically, starting from the fundamentals of signal processing to the more advanced topics of DWT and Discrete Wavelet Packet Transform.
* Written in a clear and concise manner with abundant examples, figures and detailed explanations
* Features a companion website that has several MATLAB programs for the implementation of the DWT with commonly used filters
"This well-written textbook is an introduction to the theory of discrete wavelet transform (DWT) and its applications in digital signal and image processing."
-- Prof. Dr. Manfred Tasche - Institut für Mathematik, Uni Rostock
Full review at https://zbmath.org/?q=an:06492561
Dr. D. Sundararajan, Department Head of Electrical and Electronics Engineering, Adhiyamaan College of Engineering, India.
Dr. Sundararajan obtained his PhD in Electrical Engineering at Concordia University, Montreal, Canada in 1988. As the principle inventor of the latest family of DFT algorithms, he has written three books, three Patents (which have been granted by US, Canada and Britain), and several papers in IEEE Transactions and in the Proceedings of IEEE Conference.

Chapter 2
Signals


A continuous signal is defined at all instants of time. The value of a discrete signal is defined only at discrete intervals of the independent variable (usually taken as time, even if it is not). If the interval is uniform (which is often the case), the signal is called a uniformly sampled signal. In this book, we deal only with uniformly sampled signals. In most cases, a discrete signal is derived by sampling a continuous signal. Therefore, even if the source is not a continuous signal, the term sampling interval is used. A uniformly sampled discrete signal , where is an integer and is the sampling interval, is obtained by sampling a continuous signal . That is, the independent variable in is replaced by to get . We are familiar with the discrete Fourier spectrum of a continuous periodic signal . The in the discrete independent variable is an integer, and is the fundamental frequency. Usually, the sampling interval is suppressed and the discrete signal is designated as . In actual processing of a discrete signal, its digital version, called the digital signal, obtained by quantizing its amplitude, is used. For most analytical purposes, the discrete signal is used first, and then the effect of quantization is taken into account. A two-dimensional (2-D) signal, typically an image, is a function of two independent variables in contrast to a one-dimensional (1-D) signal with a single independent variable. Figure 2.1(a) shows an arbitrary discrete signal. The discrete sinusoidal signal is shown in Figure 2.1(b). The essence of signal processing is to approximate practical signals, which have arbitrary amplitude profiles, as shown in Figure 2.1(a), and are difficult to process, by a combination of simple and well-defined signals (such as the sinusoid shown in Figure 2.1(b)) so that the design and analysis of signals and systems become simpler.

Figure 2.1 (a) An arbitrary discrete signal; (b) the discrete sinusoidal signal

2.1 Signal Classifications


2.1.1 Periodic and Aperiodic Signals


A signal is periodic, if for all values of . The smallest integer satisfying the equality is called the period of . A periodic signal repeats its values over a period indefinitely at intervals of its period. Typical examples are sinusoidal signals. A signal that is not periodic is an aperiodic signal. While most of the practical signals are aperiodic, their analysis is carried out using periodic signals.

2.1.2 Even and Odd Signals


Decomposing a signal into its components, with respect to some basis signals or some property, is the fundamental method in signal and system analysis. The decomposition of signals with respect to wavelet basis functions is the topic of this book. A basic decomposition, which is often used, is to decompose a signal into its even and odd components. A signal is odd, if

for all values of . When plotted, an odd signal is antisymmetrical about the vertical axis at the origin . A signal is even, if

for all values of . When plotted, an even signal is symmetrical about the vertical axis at the origin. An arbitrary signal can always be decomposed into its even and odd components, and , uniquely. That is,

Replacing by , we get

Adding and subtracting the last two equations, we get

Example 2.1


Find the even and odd components of the sinusoid, , shown in Figure 2.1(b).

Solution

Expressing the sinusoid in terms of its cosine and sine components, we get

Note that the even component of a sinusoid is the cosine waveform and the odd component is the sine waveform. The even and odd components are shown, respectively, in Figures 2.2(a) and (b). The decomposition can also be obtained using the defining equation. The even component is obtained as

Similarly, the odd component can also be obtained.

Figure 2.2 (a) The even component of the signal ; (b) the odd component

2.1.3 Energy Signals


The energy of a real discrete signal is defined as

An energy signal is a signal with finite energy, . The energy of the signal and is

Cumulative Energy

This is a signal measure indicting the way the energy is stored in the input signal. Let be the given signal of length . Form the new signal by taking the absolute values of and sorting them in descending order. Then, the cumulative energy of , , is defined as

Note that .

Example 2.2


Find the cumulative energy of .

Solution

Sorting the magnitude of the values of , we get as

The values of the cumulative sum of are

The cumulative energy is given by

Let the transformed representation of be

The first four of these values are obtained by taking the sum of the pairs of , and second four are obtained by taking the difference. All the values are divided by . Sorting the magnitude of the values, we get

The values of the cumulative sum of the squared values are

The cumulative energy is given by

In the case of the transformed values, the slope of the graph shown in Figure 2.3(b) is steeper than that shown in Figure 2.3(a). That is, most of the energy of the signal can be represented by fewer values.

Figure 2.3 (a) Cumulative energy of an arbitrary discrete signal; (b) cumulative energy of its transformed version

2.1.4 Causal and Noncausal Signals


Practical signals are switched on at some finite time instant, usually chosen as . Signals with are called causal signals. Signals with are called noncausal signals. The sinusoidal signal, shown in Figure 2.1(b), is a noncausal signal. Typical causal signals are the impulse and the unit-step , shown in Figure 2.4.

Figure 2.4 (a) The unit-impulse signal, ; (b) the unit-step signal

2.2 Basic Signals


Some simple and well-defined signals are used for decomposing arbitrary signals to make their representation and analysis simpler. These signals are also used to characterize the response of systems.

2.2.1 Unit-Impulse Signal


A discrete unit-impulse signal, shown in Figure 2.4(a), is defined as

It is an all-zero sequence, except that its value is one when its argument is equal to zero. In the time domain, an arbitrary signal is decomposed in terms of impulses. This is the basis of the convolution operation, which is vital in signal and system analysis and design.

Consider the product of a signal with a shifted impulse . As the impulse is nonzero only at , we get

Summing both sides with respect to , we get

The general term of the last sum, which is one of the constituent impulses of , is a shifted impulse located at with value . The summation operation sums all these impulses to form . Therefore, an arbitrary signal can be represented by the sum of scaled and shifted impulses with the value of the impulse at any being . The unit-impulse is the basis function, and is its coefficient. As the value of the sum is nonzero only at , the sum is effective only at that point. By varying the value of , we can sift out all the values of . For example, consider the signal

This signal can be expressed, in terms of impulses, as

With , for instance,

2.2.2 Unit-Step Signal


A discrete unit-step signal, shown in Figure 2.4(b), is defined as

It is an all-one sequence for positive values of its argument and is zero otherwise.

2.2.3 The Sinusoid


A continuous system is typically modeled using a differential equation, which is a linear combination of derivative terms. A sinusoid differentiated any number of times is also a sinusoid of the same frequency. The sum of two sinusoids of the same frequency but differing in amplitude and phase is also a sinusoid of the same frequency. Due to these reasons, the steady-state output of a linear time-invariant system for a sinusoidal input is also a sinusoid of the same frequency, differing only in amplitude and phase. An arbitrary signal can be decomposed into a sum of sinusoidal waveforms (Fourier analysis). Therefore, the sinusoid is the most important waveform in signal and system analysis.

There are two forms of expressions describing a sinusoid. The polar form specifies a sinusoid, in terms of its amplitude and phase, as

where , , and are, respectively, the amplitude, the angular frequency, and the phase. The amplitude is the distance of either peak of the waveform from the horizontal axis ( for the wave shown in Figure 2.1(b)). The phase angle is with respect to the reference . The peak of the cosine waveform occurs at , and its phase is zero radian. The phase of is radians. A sinusoid expressed as the sum of its cosine and sine components is its rectangular form.

The inverse relations are

A discrete sinusoid has to complete an integral number of cycles (say , where is an integer) over an integral number of sample points, called its period (denoted by , where is an integer), if it is periodic. Then, as

. Note that is the smallest integer that will make an integer. The cyclic frequency, denoted by , of a sinusoid is the number of cycles per sample and is equal to the number of cycles the sinusoid...

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