EE216B: VLSI Signal Processing. Wavelets. Prof. Dejan Marković Shortcomings of the Fourier Transform (FT)

Transcription

1 5//0 EE6B: VLSI Signal Processing Wavelets Prof. Dejan Marković Shortcomings of the Fourier Transform (FT) FT gives information about the spectral content of the signal but loses all time information FT assumes all spectral components are present at all times Time-domain representation does not provide information about spectral content 0.

2 5//0 Ambiguous Signal Representation with FT x (t) = sin(π 50t) sin(π 0t) x (t) = sin(π 50t) + sin(π 0t) 0 < t 6s 6s < t s Not all spectral components exist at all times All spectral components at all times X ( f ) X ( f ) The power spectral density (PSD) is similar, illustrating the inability of the FT to handle non-stationary spectral content 0.3 Non-stationary Signals Spectral content changes in time Cannot be supported by the Fourier-domain representation Examples: Neural action potentials, seismic signals, etc. 0.4

3 5//0 Support for Non-Stationary Signals A work-around: modify FT to allow analysis of non-stationary signals by slicing in time Short-time Fourier transform (STFT) Segment in time by applying windowing functions and analyze each segment separately Many approaches (between late 940s and early 970s) differing in the choice of windowing functions 0.5 Short-Time Fourier Transform (STFT) Time-frequency representation * jπft S( τ, f) x( t) w ( t τ) e dt x t S τ f w t τ e dτdf * ( ) (, ) ( ) j πft τ f w(t): windowing function τ: translation parameter Multiply signal by a window and then take a FT of the result (segment into stationary short-enough pieces) S(t, f ) is STFT of x(t) at frequency f and translation t Translate the window to get spectral content of signal at different times w(t t) does time-segmentation of x(t) 0.6 3

4 5//0 Good Idea, but It Doesn t Work The problem with the windowing approach is uniform resolution for all frequencies (same window for x(t)) 0.7 The Time-Frequency Resolution Problem There is a tradeoff between the resolution in time and frequency High-frequency components for a short time span Require a narrow window for time resolution But this results in wider frequency bands (cost: poor frequency resolution) Low-frequency components of longer time span Require a wider window for frequency resolution But this results in wider time bands (cost: poor time resolution) 0.8 4

5 5//0 Heisenberg's Uncertainty Principle FT is an extreme case where all time domain information is lost to get precise frequency information STFT offers fixed time/frequency resolution, which needs to be chosen according to the tradeoff we just talked about 0.9 Better Support for Non-stationary Signals? Wavelet transform Algorithm VLSI architecture 0.0 5

6 Frequency Frequency 5//0 Continuous Wavelet Transform (CWT) Addresses the resolution problem of STFT by evaluating the transform for scaled versions of the window function Time 0. CWT Algorithm Varying time and frequency resolutions are varied by using windows of different lengths * W( a, b) x( t)ψ t b dt a a Mother wavelet Time a > 0, b: scale and translation parameters 0. 6

7 Frequency Frequency 5//0 Fourier vs. Wavelet Transform Fourier basis functions Wavelet basis functions [] Time Time [] A. Graps, "An Introduction to Wavelets," IEEE Computational Science and Engineering, pp. 50-6, Summer CWT: Design Problem * W( a, b) x( t)ψ t b dt a a Mother wavelet Range of a and b? Ideally, infinitely many values of a and b would be required to fully characterize the signal Limit the range based on a signal knowledge 0.4 7

8 Amplitude 5//0 Orthonormal Wavelet Basis Wavelet representation, in general, has redundant data representation We would like to find a mother wavelet that (when translated and scaled) leads to orthonormal basis functions 0.5 Orthonormal Wavelet Basis Orthonormal wavelets have been developed Morlet, Meyer, Haar, etc. Morlet Wavelet Time 0.6 8

9 Freq. (log) Freq. (log) 5//0 Discrete Wavelet Series Discrete domain needed for digital implementation Discretize the translation and scale parameters (a, b) Example: Daubechies (a = j, b = j k) Time 0.7 Can This Be Implemented? Discretize the translation and scale parameters (a, b) Example: Daubechies (a = j, b = j k) No, because: Input signal is not yet discretized Time Number of a and b parameters still infinite 0.8 9

10 5//0 Keys to Efficient Digital Implementation Limit the number of scales used in the transform Allow support for discrete-time signals 0.9 Towards Practical Implementations How to limit the number of scales for analysis? F f( at) ω F a a ψ 4 ψ 3 ψ ψ ω Each wavelet is a like a constant Q filter 0.0 0

11 5//0 Scaling Function Scaling function spectrum (φ) cork Wavelet spectra (ψ) n+3 n+ j = n + ω n ω n ω 8 4 n j = n ω n ω 0. Mallat s Multi-Resolution Analysis Describes wavelet transform in terms of digital filtering and sampling operations Iterative use of low-pass and high-pass filters, subsequent down-sampling by x φ t h k φ t k j j ( ) j ( ) ( ) k HP g(k) γ j ψ t g k φ t k j j ( ) j ( ) ( ) k k h[ N n] ( ) g[ n] λ j h(k) LP λ j The popular form of DWT 0.

12 5//0 Mallat s Discrete Wavelet Transform Mallat showed that a subband coding structure can implement the wavelet transform The resultant filter structure is the familiar Quadrature Mirror Filter (QMF) Filter coefficients are decided based on the wavelet being used f = (π/, π) HPF g(n) x(n), f = (0, π) f = (0, π/) Level DWT Coeffs. f = (π/4, π/) LPF h(n) HPF g(n) Level DWT Coeffs. [3] f = (0, π/4) LPF h(n).. [3] S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol., no. 7, July Haar Wavelet One of the simplest and most popular wavelets It can be implemented as a filter bank shown on the previous slide g(n) and h(n) are -tap FIR filters with g(n) = [ ] h(n) = [ ] Let us look into the design of the filter in more detail 0.4

13 5//0 Haar Wavelet: Direct-Mapped Implementation Direct-mapped architecture of a stage of the Haar wavelet filter x(n) x(n) z - z - z - z - (a) Level DWT Coeffs. (b) Level DWT Coeffs. Exploit the symmetry in the coefficients to share a single multiplier for a stage 0.5 Down-Sampler Implementation Filter Output z - Down-sampled Output Interleaved Filter Output z - Down-sampled Output (a) en CTR (b) Data Valid Select en CTR CTR en Selects odd/even samples by controlling enable (en) signal of the -bit counter Interleaved down-sampler implementation Allows odd / even sample selection with different data arrival time for each channel 0.6 3

14 5//0 Polyphase Filters for DWT The wavelet filter computes outputs for each sample, half of which are discarded by the down-sampler Polyphase implementation Splits input to odd and even streams Output combined so as to generate only the output which would be needed after down-sampling Split low-pass and high-pass functions as H(z) = H e (z ) + z H o (z ) G(z) = G e (z ) + z G o (z ) Efficient computation strategy: reduces switching power by 50% 0.7 Polyphase Implementation of Haar Wavelet Consider the highlighted section of the Haar DWT filter x(n) x (n) z - z - Level DWT Coeffs. Due to down-sampling half of the computations are unnecessary Need for a realization that computes only the outputs that would be needed after down-sampling 0.8 4

15 5//0 Haar Wavelet: Sequence of Operations for h(n) Consider the computation for h(n) in the Haar wavelet filter x'(n) x'() x'() x'(3) x'(4) x'(5) x'(n ) x'(0) x'() x'() x'(3) x'(4) x'(n) + x'(n ) x'() + x'(0) x'() + x'() x'() + x'(0) x'() + x'() x'(5) + x'(4) x'() + x'(0) x'(3) + x'() x'(5) + x'(4) Computations in red are redundant Polyphase implementations Split input into odd / even streams No redundant computations performed x'(n+) x'() x'(3) x'(5) x'(n) x'(0) x'() x'(4) x'(n) + x'(n+) x'() + x'(0) x'(3) + x'() x'(5) + x'(4) 0.9 Applications of DWT Spike Sorting Useful in detection and feature extraction FBI fingerprints Efficient compression of finger prints without loosing out on information needed to distinguish between finger prints Image compression D wavelet transform provide efficient representation of images Generality of the WT lets us take a pick for the wavelet used Since a large number of wavelets exist, we can pick the right wavelet useful for the application

16 5//0 Summary FFT does not work well with non-stationary signals Wavelet is used for time-frequency analysis of non-stationary signals Multi-resolution in time and frequency is used based on orthonormal wavelet basis functions Can be implemented as a series of decimation filters Used in applications such as fingerprint recognition, image compression, neural spike sorting, etc. 0.3 References A. Graps, "An Introduction to Wavelets," IEEE Computational Science and Engineering, pp. 50-6, Summer 995. S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol., no. 7, July

17 5//0 Additional References R. Polikar, "The Story of Wavelets," in Proc. IMACS/IEEE CSCC, 999, pp T.K. Sarkar et al., "A Tutorial on Wavelets from an Electrical Engineering Perspective, Part : Discrete Wavelet Techniques," IEEE Antennas and Propagation Magazine, vol. 40, no. 5, pp , Oct V. Samar et al., "Wavelet Analysis of Neuroelectric Waveforms: A Conceptual Tutorial," Brain and Language 66, pp. 7 60, Next Monday Midterm Exam Closed Book, Closed Notes Lectures