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Fast Signal Processing Algorithms

Fast Signal Processing Algorithms. Motivations. Motivation. Why signal processing What is digital signal processing Why fast algorithms What algorithms. Why Signal Processing?. Signals are buried in noise Signals are mixed with interferences

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Fast Signal Processing Algorithms

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  1. Fast Signal Processing Algorithms Motivations

  2. Motivation • Why signal processing • What is digital signal processing • Why fast algorithms • What algorithms

  3. Why Signal Processing? • Signals are buried in noise • Signals are mixed with interferences • Useful features are to be extracted from signals (spectral analysis) • Signals are made suitable for transmission • ….

  4. Why DSP? • Digital signals can be the results of sampling continuous-time signals (such as voice signals) or discrete in nature (such as text in emails) • Signals are expressed as numbers • Signal processing can be realized by mathematical operations using computers

  5. Why DSP? • The purposes of DSP are: • Enhance the signals (noise reduction, interference elimination etc) • Present signal in a simple way (source coding) • Extract feature of signals • ….

  6. What is DSP? Output DSP DSP chips or General purpose computer Input • Input and output are discrete sequences.

  7. Why Fast Algorithms? • For achieving a specific signal processing task, we wish that: • Less computation • Less memory occupation • Short time delay

  8. What Algorithms • Two Fundamental problems in DSP • Signal analysis: spectral analysis, discrete orthogonal transforms etc • System analysis: Behavior of discrete systems with discrete input sequence – system impulse response, system functions, and input-output relationship; • We will focus on the algorithms for signal analysis

  9. What Algorithms • The focus of the subject will be: • Orthogonal transforms • DFT • DCT (Discrete Cosine Transform) • Discrete Hartkey Transforms • Discrete Hadamard transforms • Multirate techniques • Polyphase decomposition and filtering • Discrete Wavelet Transforms

  10. Fast Signal Processing AlgorithmsWeek 1 Discrete Fourier Transform

  11. Outline • What is spectrum • Spectrum analysis of analog signals • Spectral analysis of discrete sequence • Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)

  12. What is spectrum Light Spectrum: Sun light contains a range of components differing in • Color  Wavelength Frequency • The detailed structure of a beam of light in terms of those components is called the “spectrum”

  13. What is spectrum • Spectrum analysis refers to the task of working out the spectrum, that is • What color (wavelength or frequency) component does the light beam contain • How strong are those components (magnitude)

  14. Electromagnetic Spectrum

  15. Spectrum Analysis of Signals • A signals is also a “wave” and can also be considered as a combination of components with different frequencies. The detailed structure is also called the spectrum of the signal. • Therefore signal spectrum analysis refers to the task that, for a given signal: • Determineswhat frequency components it contains, and • the details of (amplitude, and phase) of those components

  16. Spectrum of analog signals • For a time-continuous signal s(t): • It can be decomposed as Fourier series if it is periodic with period T: • where • and where

  17. Spectrum of analog signals • For a time-continuous signal s(t): • If the s(t) is not periodic, Fourier Transform can be used • where • is the Fourier Transform of x(t) and is referred to as the spectrum of the signal

  18. Spectrum of analog signals • The spectrum of a periodic time-continuous signal consist of discrete frequency components, called harmonics; • The spectra of non-periodic time-continuous signals are continuous in frequency.

  19. Spectrum of Discrete signals • Let’s assume that the discrete-time signal results from periodic sampling of a continuous-time signal with sampling interval Ts: that is

  20. Spectrum of Discrete signals • Therefore the Fourier Transform of x(n) is

  21. Spectrum of Discrete signals • Considering that • is called normalized angular frequency

  22. Spectrum of Discrete signals • For simplicity the spectrum of discrete x(n) is defined as • is the normalized angular frequency. The above is also called the Discrete-time Fourier Transform (DTFT)

  23. Spectrum of Discrete signals • Assuming that sampling frequency Fs is higher than the Nyquist frequency (double of the highest frequency component), then or We have

  24. Spectrum of Discrete signals • It is not convenient to have continuous spectrum if the spectrum analysis is done by computers. For this reason we take N samples of the spectrum with their frequencies uniformly distributed between

  25. Discrete Fourier Transform • that is The above equation is referred to as the Discrete Fourier Transform (DFT) of the sequence x(n) for n=0, 1, … N-1

  26. Discrete Fourier Transform • It can be shown that The above equations are the DFT pair.

  27. DFT • The inverse transform saying that x(n) can be considered as linear combination of complex exponential sequence: X(k) is the complex amplitude of the kth complex exponential sequence. Hence X(k) give the spectrum.

  28. Discrete Fourier Transform • DFT in Matrix form:

  29. Discrete Fourier Transform • It can be shown that Recall the definition of unitary matrix, the DFT is also called an unitary transform up to the factor of N

  30. DFT– Some properties Circular time shift by L samples of x(n)

  31. DFT– Some properties Multiplication in the frequency domain. Assume that There are two signals x1(n) and x2(n) The inverse DFT of y(n) is Multiplication in the frequency domain means a circular Convolution in the time domain

  32. Spectrum of Non-periodic Signals • Let’s have a look of the DTFT again • The spectrum is determined by all the signal samples. • For example, the signal lasts 1 year, and the spectrum can only be obtained after 1 year!

  33. Spectrum of Discrete signals • There are two ways to solve the problem: • Block-based spectrum analysis • Running-based (sample based,or sliding-window based) analysis

  34. Spectrum of Discrete signals Block-based spectrum analysis: we usually take a period of observation and compute its spectrum Each observation is a time-duration limited signal

  35. Spectrum of Discrete signals • How to determine the segment size? • The resolution of DFT in spectrum (space between adjacent DFT bins) is • Larger segment size gives more accurate spectrum estimation (higher resolution)

  36. Spectrum analysis of non-periodic signals • We could increase the segment size until the DFT does not yield more details of the spectrum. • The segment size is called natural segment size.

  37. Spectrum analysis of non-periodic signals • How do we computer DFT of the segment? • Assume that a segment contains L signal samples. We would like to use N-point DFT for two reasons: • Fast computation – e.g FFT. • Change the resolution of spectrum analysis • Note that N>=L

  38. Spectrum analysis of non-periodic signals • The N-point DFT of L-point signal samples is • The computation can be performed by • padding some zeros so that the signal length is N • Compute the N-point DFT using FFT

  39. Spectrum analysis of non-periodic signals Consider the long signal again There are two sharp edges for each segment, which may results in some high frequency components which in fact do not exist

  40. Spectrum analysis of non-periodic signals A window function is applied to each segment to smooth the edges.

  41. Spectrum analysis of non-periodic signals • Many window functions have been proposed and studied: • Hamming • Hanning • Blackman • etc

  42. Fact computation of DFT’s sliding window

  43. Spectrum of Discrete signals • Assuming there are N samples within the observation period, the DTFT reduces to Note: the spectrum is still continuous in frequency

  44. DFT– zero padding We want to compute the spectrum of a signal For equidistant frequencies Where gives the resolution. N-point DFT gives the spectrum at the following frequency points

  45. DFT– zero padding If N is small, the resolution will be too low and the results will not be satisfactory. For this reason we increase the length of DFT by padding zeros to the sequence The DFT will be

  46. DFT– zero padding It is equivalent to and the resolution now is

  47. DFT– zero padding For example, we have a sequence of 128 samples, the sampling frequency is 1kHz, the 128 point DFT will give the spectrum of the signal with frequency resolution of 1/128 kHz. However, if we pad 128 zeros and do 256 point DFT, the resolution will be 1/256 kHz. Therefore zero padding is used to increase the resolution of spectrum analysis

  48. Fast Computation • We should be able to determine the block size based on • Given resolution requirement; • We wish to have efficient algorithms for compute the • block-based DFT; • We should also have efficient algorithms for sliding • window based

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