DIGITAL SIGNAL PROCESSING UNIT I

Transcription

1 DIGITAL SIGNAL PROCESSING UNIT I CONTENTS 1.1 Introduction 1.2 Introduction about signals 1.3 Signals Processing 1.4 Classification of signals 1.5 Operations performed on signals 1.6 Properties of signals 1.7 The Discrete Fourier Transform 1.8 Properties of the Discrete Fourier Transform 1.9 Linear convolution using the Discrete Fourier Transform 1.10 FFT Algorithms 1.11 Radix-2 FFT Algorithms 1.12 Decimation in Time Algorithm 1.13 Decimation in Frequency Algorithm What is DSP? DSP, or Digital Signal Processing, as the term suggests, is the processing of signals by digital means. A signal in this context can mean a number of different things. Historically the origins of signal processing are in electrical engineering, and a signal here means an electrical signal carried by a wire or telephone line, or perhaps by a radio wave. More generally, however, a signal is a stream of information representing anything from stock prices to data from a remote-sensing satellite. The term "digital" comes from "digit", meaning a number (you count with your fingers - your digits), so "digital" literally means numerical; the French word for digital is numerique. A digital signal consists of a stream of numbers, usually (but not necessarily) in binary form. The processing of a digital signal is done by performing numerical calculations. Analog and digital signals In many cases, the signal of interest is initially in the form of an analog electrical voltage or current, produced for example by a microphone or some other type of transducer. In some situations, such as the output from the readout system of a CD (compact disc) 1

2 player, the data is already in digital form. An analog signal must be converted into digital form before DSP techniques can be applied. An analog electrical voltage signal, for example, can be digitised using an electronic circuit called an analog-to-digital converter or ADC. This generates a digital output as a stream of binary numbers whose values represent the electrical voltage input to the device at each sampling instant. Signal processing Signals commonly need to be processed in a variety of ways. For example, the output signal from a transducer may well be contaminated with unwanted electrical "noise". The electrodes attached to a patient's chest when an ECG is taken measure tiny electrical voltage changes due to the activity of the heart and other muscles. The signal is often strongly affected by "mains pickup" due to electrical interference from the mains supply. Processing the signal using a filter circuit can remove or at least reduce the unwanted part of the signal. Increasingly nowadays, the filtering of signals to improve signal quality or to extract important information is done by DSP techniques rather than by analog electronics. Development of DSP The development of digital signal processing dates from the 1960's with the use of mainframe digital computers for number-crunching applications such as the Fast Fourier Transform (FFT), which allows the frequency spectrum of a signal to be computed rapidly. These techniques were not widely used at that time, because suitable computing equipment was generally available only in universities and other scientific research institutions. Digital Signal Processors (DSPs) The introduction of the microprocessor in the late 1970's and early 1980's made it possible for DSP techniques to be used in a much wider range of applications. However, generalpurpose microprocessors such as the Intel x86 family are not ideally suited to the numerically-intensive requirements of DSP, and during the 1980's the increasing importance of DSP led several major electronics manufacturers to develop Digital Signal Processor chips - specialized microprocessors with architectures designed specifically for the types of operations required in digital signal processing. (Note that the acronym DSP can variously mean Digital Signal Processing, the term used for a wide range of techniques for processing signals digitally, or Digital Signal Processor, a specialized type of microprocessor chip). Like a general-purpose microprocessor, a DSP is a programmable device, with its own native instruction code. DSP chips are capable of carrying out millions of floating point operations per second, and like their better-known general- 2

3 purpose cousins, faster and more powerful versions are continually being introduced. DSPs can also be embedded within complex "system-on-chip" devices, often containing both analog and digital circuitry. Applications of DSP DSP technology is nowadays commonplace in such devices as mobile phones, multimedia computers, video recorders, CD players, hard disc drive controllers and modems, and will soon replace analog circuitry in TV sets and telephones. An important application of DSP is in signal compression and decompression. Signal compression is used in digital cellular phones to allow a greater number of calls to be handled simultaneously within each local "cell". DSP signal compression technology allows people not only to talk to one another but also to see one another on their computer screens, using small video cameras mounted on the computer monitors, with only a conventional telephone line linking them together. In audio CD systems, DSP technology is used to perform complex error detection and correction on the raw data as it is read from the CD. Although some of the mathematical theory underlying DSP techniques, such as Fourier and Hilbert Transforms, digital filter design and signal compression, can be fairly complex, the numerical operations required actually to implement these techniques are very simple, consisting mainly of operations that could be done on a cheap four-function calculator. The architecture of a DSP chip is designed to carry out such operations incredibly fast, processing hundreds of millions of samples every second, to provide realtime performance: that is, the ability to process a signal "live" as it is sampled and then output the processed signal, for example to a loudspeaker or video display. All of the practical examples of DSP applications mentioned earlier, such as hard disc drives and mobile phones, demand real-time operation. The major electronics manufacturers have invested heavily in DSP technology. Because they now find application in mass-market products, DSP chips account for a substantial proportion of the world market for electronic devices. Sales amount to billions of dollars annually, and seem likely to continue to increase rapidly. 1.2 INTRODUCTION ABOUT SIGNALS What is a Signal? Anything which carries information is a signal. e.g. human voice, chirping of birds, smoke signals, gestures (sign language), fragrances of the flowers. 3

4 Many of our body functions are regulated by chemical signals, blind people use sense of touch. Bees communicate by their dancing pattern. Modern high speed signals are: voltage changer in a telephone wire, the electromagnetic field emanating from a transmitting antenna, variation of light intensity in an optical fiber. Thus we see that there is an almost endless variety of signals and a large number of ways in which signals are carried from on place to another place. We are all immersed in a sea of signals. All of us from the smallest living unit, a cell, to the most complex living organism (humans), receive signals all the time and continue to process them. Survival of any living organism depends upon its ability to process the signals appropriately. Signals: The Mathematical Way A signal is a real (or complex) valued function of one or more real variable(s).when the function depends on a single variable, the signal is said to be one-dimensional and when the function depends on two or more variables, the signal is said to be multidimensional. Examples of a one dimensional signal: A speech signal, daily maximum temperature, annual rainfall at a place An example of a two dimensional signal: An image is a two dimensional signal, vertical and horizontal coordinates representing the two dimensions. Four Dimensions: Our physical world is four dimensional (three spatial and one temporal). 4

5 1.3 SIGNAL PROCESSING What is Signal processing? Processing means operating in some fashion on a signal to extract some useful information e.g. we use our ears as input device and then auditory pathways in the brain to extract the information. The signal is processed by a system. In the example mentioned above the system is biological in nature. The signal processor may be an electronic system, a mechanical system or even it might be a computer program. Analog versus digital signal processing The signal processing operations involved in many applications like communication systems, control systems, instrumentation, biomedical signal processing etc can be implemented in two different ways. Analog or continuous time method Digital or discrete time method Analog signal processing Uses analog circuit elements such as resistors, capacitors, transistors, diodes etc Based on natural ability of the analog system to solve differential equations that describe a physical system The solutions are obtained in real time... Digital signal processing 5

6 The word digital in digital signal processing means that the processing is done either by a digital hardware or by a digital computer. Relies on numerical calculations The method may or may not give results in real time The advantages of digital approach over analog approach Flexibility Same hardware can be used to do various kind of signal processing operation, while in the case of analog signal processing one has to design a system for each kind of operation Repeatability: The same signal processing operation can be repeated again and again giving same results, while in analog systems there may be parameter variation due to change in temperature or supply voltage. The choice of choosing between analog or digital signal processing depends on the application. One has to compare design time, size and the cost of the implementation. 1.4 CLASSIFICATION OF SIGNALS We use the term signal to mean a real or complex valued function of real variable(s) and denote the signal by x(t) The variable t is called independent variable and the value x of t as dependent variable. When t takes a vales in a countable set the signal is called a discrete time signal. For example t ε {0, T, 2T, 3T, 4T,...} t ε {...-1, 0,1,...} t ε {1/2, 3/2, 5/2, 7/2,...} For convenience of presentation we use the notation x[n] to denote discrete time signal. When both the dependent and independent variables take values in countable sets (two sets can be quite different) the signal is called Digital Signal. 6

7 When both the dependent and independent variable take value in continous set interval, the signal is called an Analog Signal. Notation: When we write x(t) it has two meanings. One is value of x at time t and the other is the pairs (x(t), t) allowable value of t. By signal we mean the second interpretation. Notation for continuous time signal {x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in which t takes the value. Notation for discrete time signal Similarly for discrete time signal we will use the notation {x(t)}, where {x(t)} is short for {x(t), n ε I }. Note that in {x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the same signal. Some books use the notation x [.] to denote {x[n]} and x[n] to denote value of x at time n. {x(t)} refers to the whole waveform, while x[n] refers to a particular value. Most of the books do not make this distinction clean and use x[n] to denote signal and x[n 0 ] to denote a particular value. Discrete Time Signal Processing and Digital Signal Processing When we use digital computers to do processing we are doing digital signal processing. But most of the theory is for discrete time signal processing where dependent variable generally is continuous. This is because of the mathematical simplicity of discrete time signal processing. Digital Signal Processing tries to implement this as closely as possible. Thus what we study is mostly discrete time signal processing and what is really implemented is digital signal processing. Elementary Signals There are several elementary signals that occur prominently in the study of digital signals and digital signal processing. (a) UNIT SAMPLE SEQUENCE: Defined by 7

8 Graphically this is as shown below. Unit sample sequence is also known as impulse sequence. (b) UNIT STEP SEQUENCE: Defined by : Graphically this is as shown below (b) EXPONENTIALSEQUENCE: The complex exponential signal or sequence {x[n]} is defined by x[n] = C α n where C and α are, in general, complex numbers. Note that by writing α = e β, we can write the exponential sequence as x[n] = c e βn Real exponential signals: If C and are real, we can have one of the several type of behavior illustrated below 8

9 For α > 1 α < 1 magnitude of the signals grows exponentially, It is decaying exponential. 9

10 For α > 1 α < 1 all terms of {x[n]} have same sign, sign of terms in {x[n]} alternates. (d)sinusoidal SIGNAL: The sinusoidal signal {x[n]} is defined by Euler's relation allows us to relate complex exponentials and sinusoids as and The general discrete time complex exponential can be written in terms of real exponential and sinusoidal signals. Specifically if we write C and α in polar form and then Thus for α = 1, the real and imaginary parts of a complex exponential sequence are sinusoidal. α < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential, α > 1, they correspond to sinusiodal sequence multiplied by a growing exponential. In this chapter we will learn some of the operations performed on the sequences. 1.5 OPERATIONS PERFORMED ON SIGNALS a) Sequence Addition b) Scalar Multiplication c) Sequence Multiplication d) Shifting e) Reflection 10

11 a) Sequence addition: Let {x[n]} and {y[n]}be two sequences. The sequence addition is defined as term by term addition. Let {z[n]} be the resulting sequence. {z[n]} = {x[n]} + {y[n]} where each term z[n] = x[n] + y[n] We will use the following notation {x[n]} + {y[n]} = {x[n] + y[n]} b) Scalar multiplication: Let a be a scalar. We will take a to be real if we consider only the real valued signals, and take to be a complex number if we are considering complex valued sequence. Unless otherwise stated we will consider complex valued sequences. Let the resulting sequence be denoted by {w[n]} {w[n]} = a {x[n]} is defined by w[n] = ax[n] each term is multiplied by a a {w[n]} = {aw[n]} Note: If we take the set of all sequences and define these two operations as addition and scalar multiplication they satisfy all the properties of a linear vector space. c) Sequence multiplication: Let {x[n]} and {y[n]} be two sequences, and {z[n]} be resulting sequence {z[n]} = {x[n]}{y[n]} where z[n] = x[n] y[n] The notation used for this will be {x[n]} {y[n]} = {x[n] y[n]} Now we consider some operations based on independent variable n. d) Shifting: This is also known as translation. Let us shift a sequence {x[n]} by n 0 units, and the resulting sequence be {y[n]} where is the operation of shifting the sequence right by n 0 unit. The terms are defined by y[n] = x[n - n 0 ]. We will use short notation {x[n - n 0 ]} to denote shift by n 0. Figure below show some examples of shifting. {x[n]} 11

12 {x[n-2]} Consider the figure to the left. A negative value of n 0 means shift towards right. {x [n+1]} A positive value of n 0 means shift towards left. e) Reflection: Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then y[n] is defined by y[n] = x[-n] {x[n]} We will denote this by {x[-n]} 12

13 When we have complex valued signals, sometimes we reflect and do the complex conjugation, ie, y[n] is defined by y[n] = x*[-n], where * denotes complex conjugation. This sequence will be denoted by {x*[-n]}. Some of these operations commute, ie. if we apply two operations we can interchange their order and some do not commute. For example scalar multiplication and reflection commute.then v[n] = z[n] for all n. Shifting and scaling do not commute. {x[n]} {y[n]} = {x[n-1} {z[n]} = {y[-n]} 13

14 {x[n]} {w[n]} = {x[-n]} {u[n]} = {w[n-1]} We can combine many of these operations in one step, for example {y[n]} may be defined as y[n] = 2x [3-n]. 1.6 PROPERTIES OF SIGNALS a) Energy of a signal b) Power of a signal c) Periodicity of signals d) Even and Odd signals e) Periodicity property of sinusoidal signals a) Energy of a Signal: The total energy of a signal {x[n]} is defined by A signal is referred to as an energy signal, if and only if the total energy of the signal E x is finite. b) Power of a signal: If {x[n]} is a signal whose energy is not finite, we define power of the signal as A signal is referred to as a power signal if the power P x satisfies the condition An energy signal has a zero power and a power signal has infinite energy. There are signals which are neither energy signals nor power signals. For example {x[n]} defined by x[n] = n does not have finite power or energy. c) Periodic Signals: 14

15 An important class of signals that we encounter frequently is the class of periodic signals. We say that a signal {x[n] is periodic with period N, where N is a positive integer, if the signal is unchanged by the time shift of N ie., {x[n]} = {x[n + N]} or x[n] = x[ n + N ] for all n. Since {x[n]} is same as {x[n+n]}, it is also periodic so we get {x[n]} = {x[n+n]} = {x[n+n+n]} = {x[n+2n]} Generalizing this we get {x[n]} = {x[n+kn]}, where k is a positive integer. From this we see that {x[n]} is periodic with 2N, 3N,... The fundamental period N 0 is the smallest positive value N for which the signal is periodic. The signal illustrated below is periodic with fundamental period N 0 = 4 Except for a all zero signal all periodic signals have infinite energy. They may have finite power. Let {x[n]} be periodic with period N, then the power P x is given by where k is largest integer such that kn -1 M. Since the signal is periodic, sum over one period will be same for all terms. We see that k is approximately equal to M/N (it is integer part of this) and for large M we get 2M/N terms and limit 2M/(2M +1) as M goes to infinite is one we get d) Even and odd signals: A real valued signal {x[n]} is referred to as an even signal if it is identical to its time reversed counterpart ie, if {x[n]} = {x[-n]} A real signal is referred to as an odd signal if {x[n]} = {-x[-n]} An odd signal has value 0 at n = 0 as x[0] = -x[n] = - x[0] 15

16 Given any real valued signal {x[n]} we can write it as a sum of an even signal and an odd signal. Consider the signals Ev ({x[n]}) = {x e [n]} = {1/2 (x[n] + x[-n])} and Od ({x[n]}) = {x o [n]} = {1/2(x[n] -x [-n])} We can see easily that {x[n]} = {x e [n]} + {x o [n]} The signal {x e [n]} is called the even part of {x[n]}. We can verify very easily that {x e [n]} is an even signal. Similarly, {x o [n]} is called the odd part of {x[n]} and is an odd signal. When we have complex valued signals we use a slightly different terminology. A complex valued signal {x[n]} is referred to as a conjugate symmetric signal if {x[n]} = {x*[-n]} where x* refers to the complex conjugate of x. Here we do reflection and complex conjugation. If {x[n]} is real valued this is same as an even signal. A complex signal {x[n]} is referred to as a conjugate ant symmetric signal if {x[n]} = {-x*[-n]} We can express any complex valued signal as sum conjugate symmetric and conjugate ant symmetric signals. We use notation similar to above Ev({x[n]}) = {x e [n]} = {1/2(x[n] + x*[-n])} and Od ({[n]}) = {x o [n]} = {1/2(x[n] - x*[-n])} then {x[n]} = {x e [n]} + {x o [n]} We can see easily that {x e [n]} is conjugate symmetric signal and {x o [n]} is conjugate antisymmetric signal. These definitions reduce to even and odd signals in case signals takes only real values. 16

17 1.8 The Discrete Fourier Transform (DFT) Fourier Representation of Finite Duration sequence Consider the sequence such that and. Such sequences are known as finite length sequences, and N is called the length of the sequence. If a sequence has length M, we consider it to be a length N sequence where. In these cases last ( N - M ) sample values are zero. To each finite length sequence of length N we can always associate a periodic sequence defined by Note that will always be a periodic sequence with period N, whether is of finite length N or not. But when has finite length N, we can recover the sequence from by defining This is because of terms and for different values of. Recall that if n = kn + r, where then n modulo N = r, has finite length N, then there is no overlap between i.e. we add or subtract multiple of N from n until we get a number lying between 0 to N - 1. We will use ((n))n to denote n modulo N. Then for finite length sequences of length N equation (6.16) can be written as We can extract from. Thus there is one-to- one correspondance between finite length sequences of length N, and periodic sequences of period N. Given a finite length sequence we can associate a periodic sequence with it. This periodic sequence has discrete Fourier series coefficients which are also periodic with period N. From equations (6.2) and (6.3) we see that we need values of 17

18 for and for 0 = k = N - 1. Thus we define discrete Fourier transform of finite length sequence as where is DFS coefficient of associated periodic sequence. From we can get by the relation then from this we can get using synthesis equation (6.2) and finally using equation (6.17). In equations (6.2) and (6.3) summation interval is 0 to N - 1, we can write X [k ] directly in terms of x[n], and x[n] directly in terms of X[k] as For convenience of notation, we use the complex quantity DFT analysis and synthesis equations are written a follows Analysis equation: Synthesis equation:

19 If we use values of k and n outside the interval 0 to N - 1 in equation (6.8) and (6.9), then we will not get values zero, but we will get periodic repetition of and respectively. In defining DFT, we are concerned with values only in interval 0 to N - 1. Since a sequence of length M can also be considered a sequence of length, we also specify the length of the sequence by saying N-point-DFT, of sequence. Sampling of the Fourier transform: For sequence of length N, we have two kinds of representations, namely, discrete time Fourier transform and discrete Fourier transform. The DFT values can be considered as samples of (as x[n] = 0 n < 0, for n < 0, and n > N - 1) Thus is is obtained by sampling at. 1.8 PROPERTIES OF THE DISCRETE FOURIER TRANSFORM Since discrete Fourier transform is similar to the discrete Fourier series representation, the properties are similar to DFS representation. We use the notation to say that are DFT coefficient of finite length sequence. 19

20 1. Linearity If two finite length sequence have length M and N, we can consider both of them with length greater than or equal to maximum of M and N. Thus if Then. where all the DFTs are N-point DFT. 2. Circular shift of a sequence If we shift a finite length sequence When we shift it in right of length N, we face some difficulties. direction the length of the sequence will becam according to definition. Similarly if we shift it left, if may no longer be a finite length sequence as may not be zero for n < 0. Since DFT coefficients are same as DFS coefficients, we define a shift operation which looks like a shift of periodic sequence. From we get the periodic sequence defined by We can shift this sequence by m to get Now we retain the first N values of this sequence This operation is shown in figure below for m = 2, N = 5. 20

21 We can see that modulo arithmetic we have is not a shift of sequence. Using the properties of the And The shift defined in equation (6.23) is known as circular shift. This is similar to a shift of sequence in a circular register. 21

22 1. Shift property of DFT From the definition of the circular shift, it is clear that it corresponds to linear shift of the associated periodic sequence and so the shift property of the DFS coefficient will hold for the circular shift. Hence And 2. Duality We have the duality for the DFS coefficient given by, retaining one period of the sequences the duality property for the DFT coefficient will become 3. Symmetry properties We can infer all the symmetry properties of the DFT from the symmetry properties of the associated periodic sequence and retaining the first period. Thus we have And We define conjugate symmetric and anti-symmetric points in the first period 0 to N - 1 by Since the above equation similar to 22

23 and are referred to as periodic conjugate symmetric and periodic conjugate anti-symmetric parts of. In terms if these sequence the symmetric properties are 4. Circular convolution We saw that multiplication of DFS coefficients corresponds of periodic convolution of the sequence. Since DFT coefficients are DFS coefficients in the interval,, they will correspond to DFT of the sequence retained by periodically convolving associated periodic sequences and retaining their first period. Periodic convolution is given by using properties of the modulo arithmetic and then we get 23

24 The convolution defined by equation (6.28) is known as N-point-circular convolution of sequence and, where both the sequence are considered sequence of length N. From the periodic convolution property of DFS it is clear that DFT of is. If we use the notation to denote the N point circular convolution we see that In view of the duality property of the DFT we have Properties of the Discrete Fourier transform are summarized in the table Finite length sequence (length N) N-point DFT (length N) If is real sequence 24

25 1.9 Linear convolution using the Discrete Fourier Transform Output of a linear time invariant-system is obtained by linear convolution of input signal with the impulse response of the system. If we multiply DFT coefficients, and then take inverse transform we will get circular convolution. From the examples it is clear that result of circular convolution is different from the result of linear convolution of two sequences. But if we modify the two sequence appropriately we can get the result of circular convolution to be same as linear convolution. Our interests in doing linear convolution results form the fact that fast algorithms for computing DFT and IDFT are available. Here we show how we can make result of circular convolution same as that of linear convolution. If we have sequence of length L and a sequence of length M, the sequence obtained by linear convolution has length ( L + M - 1). This can be seen from the definition as x[k] = 0 for. For Now consider a sequence hence. Similarly for, so. Hence is possibly nonzero only for., DTFT is given by Writing We get 25

26 If we take we see that Comparing this with the DFT equation (6.), we see that can be seen as DFT coefficients of a sequence Obviously if has length less then or equal to N, then However, if the length of is greater than may not be equal to for all values of l. The sequence in equation (6.31) has the discrete Fourier transform The N-point DFT of sequence is where and are N-point DFTs of and respectively. The sequence resulting as the inverse DFT of is then by equation (6.32). From the circular convolution property of the DFT we have Thus, the circular convolution of two-finite length sequences can be viewed as linear convolution, followed time aliasing, defined by equation (6.32). If N is greater than or equal to ( L + M - l ), then there will be no time aliasing as the linear convolution produces a sequence of length ( L + M - l ). Thus we can use circular convolution for 26

27 linear convolution by padding sufficient number of zeros at the end of a finite length sequence. We can use DFT algorithm for calculating the circular convolution. APPLICATIONS OF DFT Linear filtering Correlation analysis Spectrum analysis 1.10 FFT ALGORITHMS A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. A DFT decomposes a sequence of values but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the obvious way, using the definition, takes O(N 2 ) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations. APPLICATIONS OF FFT FFTs are of great importance to a wide variety of applications, * Digital signal processing * Solving partial differential equations * For quick multiplication of large integers. FFT Algorithms Cooley-Tukey algorithm By far the most common FFT is the Cooley-Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size N = N 1 N 2 into many smaller DFTs of sizes N 1 and N 2, along with O(N) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966). This method (and the general idea of an FFT) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered (Heideman & Burrus, 1984) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). 27

28 The most well-known use of the Cooley-Tukey algorithm is to divide the transform into two pieces of size N / 2 at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey). These are called the radix-2 and mixed-radix cases, respectively (and other variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below RADIX-2 FFT ALGORITHMS Let us consider the computation of the N = 2 v point DFT by the divide-and conquer approach. We split the N-point data sequence into two N/2-point data sequences f 1 (n) and f 2 (n), corresponding to the even-numbered and odd-numbered samples of x(n), respectively, that is, Thus f 1 (n) and f 2 (n) are obtained by decimating x(n) by a factor of 2, and hence the resulting FFT algorithm is called a decimation-in-time algorithm. Now the N-point DFT can be expressed in terms of the DFT's of the decimated sequences as follows: But W N 2 = W N/2. With this substitution, the equation can be expressed as 28

29 where F 1 (k) and F 2 (k) are the N/2-point DFTs of the sequences f 1 (m) and f 2 (m), respectively. Since F 1 (k) and F 2 (k) are periodic, with period N/2, we have F 1 (k+n/2) = F 1 (k) and F 2 (k+n/2) = F 2 (k). In addition, the factor W N k+n/2 = -W N k. Hence the equation may be expressed as We observe that the direct computation of F 1 (k) requires (N/2) 2 complex multiplications. The same applies to the computation of F 2 (k). Furthermore, there are N/2 additional complex multiplications required to compute W N k F 2 (k). Hence the computation of X(k) requires 2(N/2) 2 + N/2 = N 2 /2 + N/2 complex multiplications. This first step results in a reduction of the number of multiplications from N 2 to N 2 /2 + N/2, which is about a factor of 2 for N large DECIMATION IN TIME ALGORITHM First step in the decimation-in-time algorithm 29

30 By computing N/4-point DFTs, we would obtain the N/2-point DFTs F 1 (k) and F 2 (k) from the relations For illustrative purposes, Figure TC.3.2 depicts the computation of N = 8 point DFT. We observe that the computation is performed in tree stages, beginning with the computations of four two-point DFTs, then two four-point DFTs, and finally, one eightpoint DFT. The combination for the smaller DFTs to form the larger DFT is illustrated in Figure TC.3.3 for N = 8. Figure TC.3.2 Three stages in the computation of an N = 8-point DFT 30

31 Figure TC.3.3 Eight-point decimation-in-time FFT algorithm. Figure TC.3.4 Basic butterfly computation in the decimation-in-time FFT algorithm. An important observation is concerned with the order of the input data sequence after it is decimated (v-1) times. For example, if we consider the case where N = 8, we know that the first decimation yeilds the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7), and 31

32 the second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input data sequence has a well-defined order as can be ascertained from observing Figure TC.3.5, which illustrates the decimation of the eight-point sequence. Figure TC.3.5 Shuffling of the data and bit reversal. 32

33 1.13 Decimation in Frequency Algorithm Another important radix-2 FFT algorithm, called the decimation-in-frequency algorithm, is obtained by using the divide-and-conquer approach. To derive the algorithm, we begin by splitting the DFT formula into two summations, one of which involves the sum over the first N/2 data points and the second sum involves the last N/2 data points. Thus we obtain Now, let us split (decimate) X(k) into the even- and odd-numbered samples. Thus we obtain where we have used the fact that W N 2 = W N/2 The computational procedure above can be repeated through decimation of the N/2- point DFTs X(2k) and X(2k+1). The entire process involves v = log 2 N stages of decimation, where each stage involves N/2 butterflies of the type shown in Figure TC.3.7. Consequently, the computation of the N-point DFT via the decimation-infrequency FFT requires (N/2)log 2 N complex multiplications and Nlog 2 N complex additions, just as in the decimation-in-time algorithm. For illustrative purposes, the eightpoint decimation-in-frequency algorithm is given in Figure TC

34 Figure TC.3.7 Basic butterfly computation in the decimation-in-frequency. 34

35 Figure TC.3.8 N = 8-piont decimation-in-frequency FFT algorithm. We observe from Figure TC.3.8 that the input data x(n) occurs in natural order, but the output DFT occurs in bit-reversed order. We also note that the computations are performed in place. However, it is possible to reconfigure the decimation-in-frequency algorithm so that the input sequence occurs in bit-reversed order while the output DFT occurs in normal order. Furthermore, if we abandon the requirement that the computations be done in place, it is also possible to have both the input data and the output DFT in normal order. 35

36 Review Questions: 1. What is a signal and what are its main types? 2. What is meant by signals processing? 3. Why DSP techniques are widely used nowadays? 4. How digital signal processors differ from microprocessors? 5. Mention few applications of DSP 6. What are the types of signal processing? 7. Which type of processing is considered as advantageous and why? 8. What are the notations used for continuous time and discrete time signals? 9. Mention the elementary signals used in the study of digital signal processing? 10. What are the operations that can be performed on signals? 11. Which of the signal operations obey commutative property? 12. How a finite duration sequence can be represented using Fourier series? 13. Write the synthesis equation and Analysis equation of DFT? 14. List the properties of DFT. 15. Does DFT obeys both linear shift and circular shift? 16. What is circular convolution of DFT? 17. What is the symmetry property of DFT? 18. How circular convolution of DFT is related with its linear convolution? 19. Mention the applications of DFT. 20. What is Fast Fourier Transform and what are its applications? 21. Which are the most preferred FFT algorithm and what is its principle? 22. What is the amount of reduction of complex multiplications by using FFT? 23. What is DIT FFT? 36

37 24. Why bit reversal of input data sequence is done in DIT FFT? 25. What are the stages of computation in 8 point DFT using DIT FFT? 26. What is DIF FFT? 27.. What are the stages of computation in 8 point DFT using DIF FFT? 28. What is the order of input sequence and output sequence for DIF FFT? 37