EECE 491 Lecture 11, Spring 2016

Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 1 EECE 491: Discrete-­‐time Signal Processing Mohammad M. Mansour Dept. of Electrical and Compute Engineering American University of Beirut Lecture 11: Multirate Signal Processing Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 2 Announcements § Reading – O&S • Chapter 4 Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 3 Outline § Filtering and compression/expansion § Polyphase decompositions § Multirate filter banks Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 4 Sample-­‐Rate Conversion § We can change the sample rate of D.T. signal by a combination of interpolation and decimation – To obtain a new sampling rate of 1.01T, we can first interpolate by L = 100, then decimate by M = 101. – These large intermediate changes in sampling rate would require large amounts of computations. § Can take advantages of techniques broadly known as multirate signal processing – They utilize upsampling, downsampling, compressors, and expanders to increase efficiency of SP systems – Applications besides sample rate conversion: • A/D and D/A systems that exploit oversampling and noise shaping • Filter banks for the analysis and processing of signals § We will explore two basic results of multirate signal processing 1. Interchanging of filtering and downsampling/upsampling operations 2. Polyphase decomposition Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 5 Compressor/Expander § For a compressor-­‐by-­‐M with input x[n] and output xd[n], we have § For an expander-­‐by-­‐L with input x[n] and output xe[n], we have ( ) ( ) ( ) 1 2 / 0 1 M j i M j d i X e X e M ω π ω − − = = ∑ ( ) ( ) j j L Xe e X e ω ω = Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 6 Example § Consider two systems A and B § Do the two systems have the same input-­‐output relationship? § If x[n] is clocked at 1 sample/second, how many multiplications per second are required in System A? § How many in system B? YES N multiplications per second N/M multiplications per second N Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 7 Noble Identity I: Interchange of Filtering with Compressor § Prove the following two systems are equivalent § For System B, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 / 0 1 2 2 / 0 1 2 / 0 1 1 1 M j i M j b i M j i j i M i M j i M j i j j a Y e X e M H e X e M H e X e M H e X e ω π ω ω π ω π ω π ω ω ω − − = − − − = − − = = ∑ = ∑ = ∑ = ( ) ( ) ( ) j j M j Xb e H e X e ω ω ω = System A System B Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 8 Noble Identity I: Interchange of Filtering with Compressor § What is the relationship between H(z) and H(zM)? – Example: Let H(z) = 1 + 2z–1 + 3z–2 . – For M = 3, we have H(z3) = 1 + 2z–3 + 3z–6 – Hence the filter response H(zM) is an expanded version of H(z) by a factor of M. – For H(zM), we have ℎ 𝑛 = 0, ∀𝑛 ≠ 𝑘𝑚, any integer k. § Therefore, identity I says that: – Moving an LTI filter which is at the output of a compressor-­‐by-­‐M to the compressor’s input expands the filter’s response by M. – The converse is not always true – Moving an LTI filter H which is at the input of a compressor-­‐by-­‐M to its output compresses the filter’s response by M only if the LTI filter can be represented as H(zM) using integer powers of zM. Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 9 Example: Intuitive Explanation of Identity I When a signal x[n] is processed by filtering with some H(zM) (which implies ℎ 𝑛 = 0, ∀𝑛 ≠ 𝑘𝑚, any integer k) and compressing by a factor of M, each output sample y[n] depends only on every Mth input sample. It therefore stands to reason that an equivalent input-­‐output relationship can be achieved by first compressing, then filtering. Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 10 Noble Identity II: Interchange of Filtering with Expander § Prove the following two systems are equivalent § For System A, we have ( ) ( ) ( ) ( ) j j L a j L j L Y e X e H e X e ω ω ω ω = = ( ) ( ) j L j b H e X e ω ω = which corresponds to system B System A System B Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 11 Multistage Decimation § When decimation ratios are large, need to use long filters to achieve adequate approximations to the required lowpass filters – Significant reduction in computations can be achieved by using multistage decimation (a)  2-­stage  decimation system  with  overall decimation  ratio  M  =  M1M2. LPF  with �𝜔+ = 𝜋/𝑀/ [ ] [ ] [ ] [ ] 1 2 1 k h n h n h k n kM δ ∞ =−∞ = ∗ − ∑ ( ) ( ) ( 1 ) 1 2 M H z H z H z = (b)  Modification  of  (a)  using downsampling  identity  I (c)  Equivalent  one-­stage decimation. LPF  with �𝜔+ = 𝜋/𝑀0 Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 12 Multistage Interpolation § Similar concept holds for interpolation Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 13 Polyphase Decompositions Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 14 Polyphase Decompositions § Polyphase decomposition of h[n]: – Represent h[n] as a superposition of M subsequences – Each subsequence consists of every Mth value of successively delayed versions of h[n] – These subsequences ek[n] are called the kth polyphase components of h[n] § Example Q: Given the polyphase components ek[n], how might the original filter h[n] be implemented? [ ] [ ] multiple of 0 otherwise k h n k n M h n ⎧ + = ⎨ ⎩ h0 [ ] n [ ] 1h n h2 [ ] n [ ] [ ] 1 0 M k k h n h n k − = = − ∑ Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 15 Generating the Polyphase Components of h[n] [ ] [ ] [ ] k k e n h nM k h nM = + = [ ] [ ] multiple of 0 otherwise k h n k n M h n ⎧ + = ⎨ ⎩ Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 16 Constructing h[n] from its Polyphase Components Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 17 Polyphase Decomposition of h[n] using Polyphase Components E0 ( ) z E1 ( ) z E2 ( ) z EM 1 ( ) − z ( ) 1 M EM z − ( ) 2 M E z ( ) 1 M E z ( ) 0 M E z ( ) ( ) 1 0 M M k k k H z E z z − − = = ∑ Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 18 Polyphase Decomposition of h[n] using Chained Delays (cont’d) E0 ( ) z E1 ( ) z E2 ( ) z EM 1 ( ) z − ( ) 0 M E z ( ) 1 M E z ( ) 2 M E z ( ) 1 M EM z − Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 19 Realization Structure Based on Polyphase Decomposition ( ) ( ) 1 0 M M k k k H z E z z − − = = ∑ Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 20 Polyphase Implementation of Decimation Filters N-­‐point filter N MULTs, N – 1 ADDs 1 sample / sec ( ) ( ) 1 0 M M k k k H z E z z − − = = ∑ 1 sample / sec 1 sample / M sec N/M-­‐point filters Each filter requires: • 1/M * (N/M) MULTs • 1/M*(N/M – 1) ADDs Total: • (N/M) MULTs • (N/M – 1) + M – 1 ADDs Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 21 Polyphase Implementation of Interpolation Filters Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 22 Polyphase Implementation of Filters § For both decimation and interpolation filters – Gains in computational efficiency result from rearranging operations so that filtering is done at the low sampling rate § Combinations of interpolation and decimation systems for non-­‐integerrate changes lead to significant savings when high intermediate rates are required Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 23 Multirate Filter Banks § Polyphase structures for decimation/interpolation are used in filter banks for analysis/synthesis of audio and speech signals § Two-­‐channel analysis and synthesis filter bank LPF �c=�/2 HPF, �c=�/2 [ ] [ ] ( ) ( ) ( ) 1 0 1 0 j n j j h n e h n H e H e π ω π ω − = = LPF �c=�/2 HPF, �c=�/2 Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 24 Two-­‐Channel Analysis and Synthesis Filter Bank § Frequency domain relations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 /2 2 /2 /2 /2 0 0 0 1 2 j j j j j V e H e X e H e X e ω π ω π ω ω ω − − = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 /2 2 /2 /2 /2 1 1 1 1 2 j j j j j V e H e X e H e X e ω π ω π ω ω ω − − = + ( ) ( ) ( ) 2 0 0 0 j j j Y e G e V e ω ω ω = ( ) ( ) ( ) 2 1 1 1 j j j Y e G e V e ω ω ω = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 2 0 0 1 1 0 0 0 1 1 1 1 2 1 2 j j j j j j j j j j j j j j j j j Y e Y e Y e G e V e G e V e G e H e X e H e X e G e H e X e H e X e ω ω ω ω ω ω ω ω π ω π ω ω ω ω π ω π ω ω ω − − − − = + = + = + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 1 0 0 1 1 1 2 1 2 j j j j j j j j j j j Y e G e H e G e H e X e G e H e G e H e X e ω ω ω ω ω ω ω π ω π ω π ω ω − − − ⎡ ⎤ = + ⎣ ⎦ ⎡ ⎤ + + ⎢ ⎥ ⎣ ⎦ Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 25 Two-­‐Channel Analysis and Synthesis Filter Bank § If analysis and synthesis filters are ideal, then get perfect reconstruction: § If filters are non-­‐ideal, can still obtain perfect or near perfect reconstruction if § This condition is called alias cancellation condition. § One set of conditions that satisfy the above equation is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 1 0 0 1 1 1 2 1 2 j j j j j j j j j j j Y e G e H e G e H e X e G e H e G e H e X e ω ω ω ω ω ω ω π ω π ω π ω ω − − − ⎡ ⎤ = + ⎣ ⎦ ⎡ ⎤ + + ⎢ ⎥ ⎣ ⎦ ( ) ( ) j j Y e X e ω ω = Can potentially introduce aliasing ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 1 0 j j j j G e H e G e H e ω π ω π ω ω − − + = [ ] [ ] ( ) ( ) ( ) [ ] [ ] ( ) ( ) [ ] [ ] ( ) ( ) ( ) 1 0 1 0 0 0 0 0 1 0 1 0 2 2 2 2 j j n j j j j j h n e h n H e H e g n h n G e H e g n h n G e H e ω π π ω ω ω ω π ω − − = ⇒ = = ⇒ = = − ⇒ = − h0[n] and h1[n] are Quadrature Mirror Filters They impose symmetry about �=�/2 Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 26 Two-­‐Channel Analysis and Synthesis Filter Bank § Substituting back, we obtain: § Perfect reconstruction è § It can be shown that the only computationally realizable filters satisfying (**) are: § Such filters cannot provide sharp frequency selective properties needed in speech and audio applications § But to illustrate the concept, consider 2-­‐point moving average filter § Obtain: ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 0 j j j j Y e H e H e X e ω π ω ω ω − ⎡ ⎤ = − ∗∗ ⎢ ⎥ ⎣ ⎦ ( ) ( ) ( ) 2 2 0 0 j j j M H e H e e ω π ω ω − − − = (allow a phase shift of M) [ ] [ ] [ ] 0 0 0 1 1 0 1 0 1 0 1 2 2 1 , arbitrary integers 1 , satisfy 4 h n c n n c n n n n c c c c δ δ = − + − − = [ ] [ ] [ ] ( ) ( ) ( ) /2 0 0 1 1 cos / 2 2 j j h n n c n H e e ω ω δ δ ω − = + − = ( ) ( ) j j j Y e e X e ω ω ω = − Prof. M. Mansour EECE 491: Discrete-­‐time Signal Processing 27 Two-­‐Channel Analysis and Synthesis Filter Bank § Polyphase techniques can be used to save computations in the implementation of analysis/synthesis filter banks [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 00 0 01 0 2 10 1 0 00 2 11 0 0 01 2 2 1 2 2 2 1 2 1 j n j n j e n h n e n h n e n h n e h n e n e n h n e e h n e n π π π = = + = = = = + = + = −