DISCRETE FOURIER TRANSFORM UNIT-I DISCRETE FOURIER TRANSFORM

Syllabus Discrete Signals and Systems- A Review Introduction to DFT Properties of DFT Circular Convolution Filtering methods based on DFT FFT Algorithms Decimation in time Algorithms, Decimation in frequency Algorithms Use of FFT in Linear Filtering.

Signal Processing Humans are the most advanced signal processors speech and pattern recognition, speech synthesis,… We encounter many types of signals in various applications Electrical signals: voltage, current, magnetic and electric fields,… Mechanical signals: velocity, force, displacement,… Acoustic signals: sound, vibration,… Other signals: pressure, temperature,… Most real-world signals are analog They are continuous in time and amplitude Convert to voltage or currents using sensors and transducers Analog circuits process these signals using Resistors, Capacitors, Inductors, Amplifiers,… Analog signal processing examples Audio processing in FM radios Video processing in traditional TV sets

Limitations of Analog Signal Processing Accuracy limitations due to Component tolerances Undesired nonlinearities Limited repeatability due to Tolerances Changes in environmental conditions Temperature Vibration Sensitivity to electrical noise Limited dynamic range for voltage and currents Inflexibility to changes Difficulty of implementing certain operations Nonlinear operations Time-varying operations Difficulty of storing information

Digital Signal Processing Represent signals by a sequence of numbers Sampling or analog-to-digital conversions Perform processing on these numbers with a digital processor Digital signal processing Reconstruct analog signal from processed numbers Reconstruction or digital-to-analog conversion A/D DSP D/A analog signal digital signal Analog input – analog output Digital recording of music Analog input – digital output Touch tone phone dialing Digital input – analog output Text to speech Digital input – digital output Compression of a file on computer

Pros and Cons of Digital Signal Processing Accuracy can be controlled by choosing word length Repeatable Sensitivity to electrical noise is minimal Dynamic range can be controlled using floating point numbers Flexibility can be achieved with software implementations Non-linear and time-varying operations are easier to implement Digital storage is cheap Digital information can be encrypted for security Price/performance and reduced time-to-market Cons Sampling causes loss of information A/D and D/A requires mixed-signal hardware Limited speed of processors Quantization and round-off errors

Discrete Signals and Systems

Discrete-Time Signals: Sequences Discrete-time signals are represented by sequence of numbers The nth number in the sequence is represented with x[n] Often times sequences are obtained by sampling of continuous-time signals In this case x[n] is value of the analog signal at xc(nT) Where T is the sampling period 20 40 60 80 100 -10 10 t (ms) 30 50 n (samples)

Basic Sequences and Operations Delaying (Shifting) a sequence Unit sample (impulse) sequence Unit step sequence Exponential sequences -10 -5 5 10 0.5 1 1.5

Sinusoidal Sequences Important class of sequences An exponential sequence with complex x[n] is a sum of weighted sinusoids Different from continuous-time, discrete-time sinusoids Have ambiguity of 2k in frequency Are not necessary periodic with 2/o

Discrete-Time Systems Discrete-Time Sequence is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n] Example Discrete-Time Systems Moving (Running) Average Maximum Ideal Delay System x[n] T{.} y[n]

Memoryless System Memoryless System A system is memoryless if the output y[n] at every value of n depends only on the input x[n] at the same value of n Example Memoryless Systems Square Sign Counter Example Ideal Delay System

Linear Systems Linear System: A system is linear if and only if Examples Ideal Delay System

Time-Invariant Systems Time-Invariant (shift-invariant) Systems A time shift at the input causes corresponding time-shift at output Example Square Counter Example Compressor System

Causal System Causality Examples Counter Example A system is causal it’s output is a function of only the current and previous samples Examples Backward Difference Counter Example Forward Difference

Stable System Stability (in the sense of bounded-input bounded-output BIBO) A system is stable if and only if every bounded input produces a bounded output Example Square Counter Example Log

DFS,DTFT & DFT

Fourier Transform of continuous time signals

Linear Time Invariant (LTI) Systems and z-Transform If the system is LTI we compute the output with the convolution: If the impulse response has a finite duration, the system is called FIR (Finite Impulse Response):

Z-Transform Facts: Frequency Response of a filter:

Frequency-Domain Properties Time Continuous Discrete Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform DFT Duration Finite Infinite Discrete-Time and z-Transform & Periodic (2) Discrete & Periodic

Signal Processing Methods Periodicity Periodic Aperiodic Fourier Series Continuous-Time Fourier Transform Time Continuous Discrete DFS Discrete-Time Fourier Transform and z-Transform Duration Finite Infinite

The Four Fourier Methods 28

Relations Among Fourier Methods 29

The Family of Fourier Transform Aperiodic-Continuous－Fourier Transform

The Family of Fourier Transform Periodic-Continuous－Fourier Series

The Family of Fourier Transform Periodic-Discrete－DFS (DFT)

The Discrete Fourier Transform (DFT) The DFS provided us a mechanism for numerically computing the discrete-time Fourier transform. But most of the signals in practice are not periodic. They are likely to be of finite length. Theoretically, we can take care of this problem by defining a periodic signal whose primary shape is that of the finite length signal and then using the DFS on this periodic signal. Practically, we define a new transform called the Discrete Fourier Transform, which is the primary period of the DFS. This DFT is the ultimate numerically computable Fourier transform for arbitrary finite length sequences. Introduction 为了计算有限长序列的傅氏变换，可定义一个周期性序列，有限长序列正好是该周期序列的一个周期，因此就可用DFS求其频谱。 33

The Discrete Fourier Transform (DFT) Finite-length sequence & periodic sequence Finite-length sequence that has N samples periodic sequence with the period of N Window operation Periodic extension

The Family of Fourier Transform Aperiodic-Discrete－DTFT

DFT and IDFT

IDFT

The Discrete Fourier Transform (DFT) The definition of DFT

Twiddle factor

Relationship of DFT to other transforms

Problems on DFT

Example of DFT Find X[k] We know k=1,.., 7; N=8

Example of DFT

Example of DFT Polar plot for Time shift Property of DFT

Example of DFT

Example of DFT

Using the shift property! Example of DFT Summation for X[k] Using the shift property!

Using the shift property! Example of DFT Summation for X[k] Using the shift property!

Example of IDFT Remember:

Example of IDFT Remember:

Problems on DFT

Problems on DFT

Circular shift of a sequence

The Properties of DFT Linearity Circular shift of a sequence N3-point DFT, N3=max(N1,N2) Circular shift of a sequence Circular shift in the frequency domain

The Properties of DFT The sum of a sequence The first sample of sequence

The Properties of DFT Circular convolution N N N Multiplication

The Properties of DFT Circular convolution N N N Multiplication

The Properties of DFT Circular correlation Linear correlation

The Properties of DFT

The Properties of DFT Parseval’s theorem

The Properties of DFT Conjugate symmetry properties of DFT and Let be a N-point sequence x(-n)=x(N-n); 将x(n)看成是x~(n)的一个周期 在这里讲为何将x*(-n)的周期延拓写成x*((N-n))N期,主要从取模运算的方便性考虑 It can be proved that 69

The Properties of DFT Circular conjugate symmetric component Circular conjugate antisymmetric component

The Properties of DFT 71 对xep(n)的计算除了给出的画图方法之外，用直接计算的方法讲一遍。 举一个复数序列求圆周共轭对称分量的例子 x(n)={2+j1, 3+j2, 4-j3}, x*(n)={2-j1, 3-j2, 4+j3} x*((N-n))NRN(n) ={2-j1, 4+j3,3-j2} xep(n) ={2, 7/2+j5/2, 7/2-j5/2} 验证：xep*((N-n))NRN(n) ={2, 7/2+j5/2, 7/2-j5/2}=xep(n) 71

The Properties of DFT and

The Properties of DFT 73 由Xep(k)和Xop(k)的定义即可得此公式 这里的偶对称指得是圆周偶对称。 73

The Properties of DFT Circular even sequences Circular odd sequences 实数偶对称－－实数偶对称 实数奇对称－－虚数奇对称 证明 74

Circular convolution in Time domain

Circular convolution

Circular convolution-Example

One sequence is distributed clockwise and x1(n)=(1,2,3) and x2(n)=(4,5,6). One sequence is distributed clockwise and the other counterclockwise and the shift of the inner circle is clockwise. 1 1 * * 6 4 + + + + =6+10+12=28 =4+12+15=31 4 5 5 6 * * * 3 2 * 3 2 + + 1st term 2st term 85

x1(n)* x2(n)=(1,2,3)*(4,5,6)=(18,31,31). We have the same symmetry as before 3rd term 1 * 5 + + =5+8+18=31 6 4 * * 3 2 + 86

Cirular convolution-Matrix method

Circular convolution property

Use of DFT in linear filtering DFT- based filtering always performs circular convolution. But by zero padding adequetly,Circular convoltion can yield the same result. where n=0.....(L+M-1)-1

Linear convolution using DFT

LINEAR & CIRCULAR CONVOLUTION (1)calculate N point circular convolution by linear convolution (2) calculate linear convolution by circular convolution (3) calculate linear convolution by DFT

Example:Cir.conv

Circular conv

FAST FOURIER TRANSFORM

Discrete Fourier Transform The DFT pair was given as Baseline for computational complexity: Each DFT coefficient requires N complex multiplications N-1 complex additions All N DFT coefficients require N2 complex multiplications N(N-1) complex additions Complexity in terms of real operations 4N2 real multiplications 2N(N-1) real additions Most fast methods are based on symmetry properties Conjugate symmetry Periodicity in n and k

Properties-Example

Decimation-In-Time FFT Algorithms Makes use of both symmetry and periodicity Consider special case of N an integer power of 2 Separate x[n] into two sequence of length N/2 Even indexed samples in the first sequence Odd indexed samples in the other sequence Substitute variables n=2r for n even and n=2r+1 for odd G[k] and H[k] are the N/2-point DFT’s of each subsequence

Decimation In Time 8-point DFT example using decimation-in-time Two N/2-point DFTs 2(N/2)2 complex multiplications 2(N/2)2 complex additions Combining the DFT outputs N complex multiplications N complex additions Total complexity N2/2+N complex multiplications N2/2+N complex additions More efficient than direct DFT Repeat same process Divide N/2-point DFTs into Two N/4-point DFTs Combine outputs

DIT-FFT

Decimation In Time Cont’d After two steps of decimation in time Repeat until we’re left with two-point DFT’s

Decimation-In-Time FFT Algorithm Final flow graph for 8-point decimation in time Complexity: Nlog2N complex multiplications and additions

Butterfly Computation Flow graph constitutes of butterflies We can implement each butterfly with one multiplication Final complexity for decimation-in-time FFT (N/2)log2N complex multiplications and additions

Decimation-in-time flow graphs require two sets of registers In-Place Computation Decimation-in-time flow graphs require two sets of registers Input and output for each stage Note the arrangement of the input indices Bit reversed indexing

FFT Implementation Decimation in time FFT: Number of stages = log2N Example: 8 point FFT (1)Number of stages: Nstages = 3 (2)Blocks/stage: Stage 1: Nblocks = 4 Stage 2: Nblocks = 2 Stage 3: Nblocks = 1 (3)B’flies/block: Stage 1: Nbtf = 1 Stage 2: Nbtf = 2 Stage 3: Nbtf = 2 W0 -1 W0 -1 W0 -1 W2 -1 W0 -1 W0 -1 W1 -1 W0 W0 -1 W2 -1 W0 -1 W2 -1 W3 -1 Decimation in time FFT: Number of stages = log2N Number of blocks/stage = N/2stage Number of butterflies/block = 2stage-1

FFT Implementation Start Index Input Index 1 2 4 Twiddle Factor Index -1 W2 W1 W3 Stage 2 Stage 3 Stage 1 Start Index Input Index 1 2 4 Twiddle Factor Index N/2 = 4 4 /2 = 2 2 /2 = 1 Indicies Used W0 W2 W1 W3

Decimation-in-time FFT algorithm Most conveniently illustrated by considering the special case of N an integer power of 2, i.e, N=2v. Since N is an even integer, we can consider computing X[k] by separating x[n] into two (N/2)-point sequence consisting of the even numbered point in x[n] and the odd-numbered points in x[n]. or, with the substitution of variable n=2r for n even and n=2r+1 for n odd

That is, WN2 is the root of the equation WN/2=1 Consequently, Since That is, WN2 is the root of the equation WN/2=1 Consequently, Both G[k] and H[k] can be computed by (N/2)-point DFT, where G[k] is the (N/2)-point DFT of the even numbered points of the original sequence and the second being the (N/2)-point DFT of the odd-numbered point of the original sequence. Although the index ranges over N values, k = 0, 1, …, N-1, each of the sums must be computed only for k between 0 and (N/2)-1, since G[k] and H[k] are each periodic in k with period N/2.

Decomposing N-point DFT into two (N/2)-point DFT for the case of N=8

We can further decompose the (N/2)-point DFT into two (N/4)-point DFTs We can further decompose the (N/2)-point DFT into two (N/4)-point DFTs. For example, the upper half of the previous diagram can be decomposed as

Hence, the 8-point DFT can be obtained by the following diagram with four 2-point DFTs.

Finally, each 2-point DFT can be implemented by the following signal-flow graph, where no multiplications are needed. Flow graph of a 2-point DFT

Flow graph of complete decimation-in-time decomposition of an 8-point DFT.

The butterfly computation can be simplified as follows: In each stage of the decimation-in-time FFT algorithm, there are a basic structure called the butterfly computation: The butterfly computation can be simplified as follows: Flow graph of a basic butterfly computation in FFT. Simplified butterfly computation.

Flow graph of 8-point FFT using the simplified butterfly computation

In the above, we have introduced the decimation-in-time algorithm of FFT. Here, we assume that N is the power of 2. For N=2v, it requires v=log2N stages of computation. The number of complex multiplications and additions required was N+N+…N = Nv = N log2N. When N is not the power of 2, we can apply the same principle that were applied in the power-of-2 case when N is a composite integer. For example, if N=RQ, it is possible to express an N-point DFT as either the sum of R Q-point DFTs or as the sum of Q R-point DFTs. In practice, by zero-padding a sequence into an N-point sequence with N=2v, we can choose the nearest power-of-two FFT algorithm for implementing a DFT. The FFT algorithm of power-of-two is also called the Cooley-Tukey algorithm since it was first proposed by them. For short-length sequence, Goertzel algorithm might be more efficient.

Decimation-In-Frequency FFT Algorithm Final flow graph for 8-point decimation in frequency

Decimation-In-Frequency FFT Algorithm The DFT equation Split the DFT equation into even and odd frequency indexes Substitute variables to get Similarly for odd-numbered frequencies

Decimation in Frequency algorithm

Decimation in Frequency

DIF-FFT ALGORITHM

DIF-FFT ALGORITHM

Example Using decimation-in-time FFT algorithm compute DFT of the sequence {-1 –1 –1 –1 1 1 1 1} Solution: Twiddle factors are

Solution and signal flow graph of the example

IDFT Using DIF-FFT