REAL TIME DIGITAL SIGNAL PROCESSING
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1 REAL TIME DIGITAL SIGNAL PROCESSING
2 Digital Filters FIR and IIR. Design parameters. Implementation types. Constraints.
3 Filters: General classification
4 Filters: General classification SW Smith, The Scientist and Engineer s guide to DSP. California Tech. Pub. 997.
5 Filters: Frequency domain features Fast roll-off, flat passband and high attenuation in stopband are desirable features. They can t be achieved at the same time, so the design procedure is a trade off between these features.
6 Filters: Time domain features Step response evidence the most important time domain features. In this case the trade off is between fast step response and overshoot. Overshoot is an undesirable distortion and must be avoided. Simetry is a fingerprint of a linear phase response system (FIR).
7 Filters: Phase response The phase response of a filter gives the radian phase shift experienced by each sinusoidal component of the input signal. Sometimes it is more meaningful to consider phase delay
8 Filters: Group and Phase Delay
9 Filters: Group and Phase Delay Group Delay Normalized Frequency: Group delay: H(w) Group del lay (in samples) Phase Delay Normalized Frequency: Phase Delay: Normalized Frequency ( π rad/sample) Pole/Zero Plot Phase Delay (samples) Imaginary Part Normalized Frequency ( π rad/sample) Real Part
10 Filters: Group Delay W 0. W W 0.5 W Gd(0,) 343 samples Input Output
11 Filters: Phase Delay Pd(0,) 3 samples
12 Digital Filters: Design steps Requirement specifications. Coefficients calculation. Filter realization: Structure. Wordlength effects. Software or Hardware implementation. EC Ifeachor, BW Jervis. Digital Signal Processing. A Practical approach. Second Edition. Prentice Hall.
13 Digital Filters: Specifications H ( ω ) δ ( ) +, 0 p H ω δ p ω ω p + δ p δ p A p Ideal filter Actual filter A p + δ p 20 log0 db δ p A s H( ω) δ s, ω ω π s A p : Ripple in band (db) A s : Band-stop attenuation (db) π-ω s : Band-stop start/end ω p -ω s : Band-pass start/end δ s 0 Passband ω p ω c ω s Transition band Stopband π ω A s 20 log δ db 0 s
14 Finite Impulse Response FIR
15 FIR filters: General characteristics FIR digital filters use only current and past input samples. Require no feedbac. Their transfer function has zeros around the Z plane and poles at the origin, so FIR filters are always stable. They can easily be designed to be linear phase and constant delay by maing the coefficient sequence symmetric. Finite duration of nonzero input values / Finite duration of nonzero output values. Finite Impulse Response.
16 FIR filters: General characteristics Being x(n) and h(n) are arrays of numbers. If we want to compute y(n) we have to multiply and sum the last M samples, being M the length of h(n). As you can see, any linear system uses multiplications, accumulations (sums), and loops intensively. UTN - FRBA 20
17 FIR filters: Perfect linear phase pd delay gd Θ f M 2 ( f ) [ samples] Zero phase filters are a particular case of linear phase with zero delay. gd 0 gd 25 delay 25
18 Linear phase FIR filter Type I: h(n) h(n-n) with even N. Type II: h(n) h(n-n) with odd N. Type III: h(n) -h(n-n) with even N. Type IV: h(n) -h(n-n) with odd N.
19 Type I Even order - Even symmetric Pole/Zero Plot 0.5 Impulse Response Imaginary Part Amplitude Real Part Magnitude (db) and Phase Responses Time (useconds) Magnitude (db) Phase (radians) Frequency (Hz)
20 Type II Odd order Even symmetric Pole/Zero Plot 0.6 Impulse Response Imaginary Part Amplitude Real Part Magnitude (db) and Phase Responses Time (useconds) We can t design high pass filters (HPF) Magnitude (db) Phase (radians) Frequency (Hz)
21 Type III Even order - Odd symmetric Pole/Zero Plot Impulse Response Imaginary Part 0 20 Amplitude Real Part Samples Magnitude (db) and Phase Responses We can t design low and high pass filters (LPF and HPF) Magnitude (db) Phase (radians) Normalized Frequency ( π rad/sample)
22 Type IV Odd order Odd symmetric Pole/Zero Plot Impulse Response Imaginary Part 0 2 Amplitude Real Part Time (useconds) Magnitude (db) and Phase Responses We can t design low pass filters (LPF) Magnitude (db) Phase (radians) Frequency (Hz)
23 Minimun phase FIR filter Pole/Zero Plot Pole/Zero Plot.5 Imaginary Part Imaginary Part Real Part Magnitude (db) and Phase Responses Real Part Magnitude (db) and Phase Responses Magnitude (db) Phase (radians) Magnitude (db) Phase (radians) Frequency (Hz) Frequency (Hz)
24 Minimun phase FIR filter Impulse Response Impulse Response Amplitude 0 Amplitude Time (mseconds) Group Delay Time (mseconds) Group Delay Group delay (in samples) Group delay (in samples) Frequency (Hz) Frequency (Hz)
25 Minimum-Phase vs Linear Phase L-P FIR filter has zeros inside, on, and outside the unit circle. M-P FIR filter has all of its zeros on or within the unit circle. Given optimal M-P and L-P digital FIR filters that meet the same magnitude specification, the M-P filter would have a reduced filter length of typically one half to three fourths of the L-P filter length, and Minimum group delay, where energy would be concentrated in the low-delay instead of the medium-delay coefficients M-P filters can simultaneously meet constraints on delay and magnitude response while generally requiring fewer computations and less memory than L-P filters.
26 FIR filters: Coefficients calculation Moving average (MA) Windowed Sinc filters Remez exchange (Pars McClellan) method
27 Moving average filters Each output will be the average of the last M samples of the input. This ind of filters is very easy to implement, but it has very poor roll-off and attenuation in stopband. It is mainly used for smoothing purposes ] / [ ] [ ; ] [ ] [ M j M j M j n x M n y j n x M n y
28 MA filters: applications Improvements can be achieved if we increase M or if the same filter is passed multiple times.
29 MA filters: Recursive Implementation The cost of implementing a MA filter is M sums and multiplication. This can be improved if a recursive approach is taen. So any MA filter can be implemented by 2 sums and one accumulator. Keep in mind that a higher M implies a higher delay.
30 MA Filter: Example.5 input filter output MA filter output FIR filter MA filter: M50, An0%, FIR filter: Equiripple order 39
31 Windowed Sinc FIR filters A sinc of infinite duration is the time domain equivalent of an ideal filter response. As it is impossible to store this signal, it is truncated to M samples. This truncation causes a severe degradation to the ideal frequency response.
32 Windowed Sinc FIR filters In order to obtain better filter features, the truncated sinc can be multiplied by a window function. These windows achieve either better smoothness, roll-off or stopband attenuation than the sinc. For this purposes, many windows were designed. x
33 Windowed Sinc FIR filters: Design Windowed Sinc filters are very easy to design. We only have to multiply a window formula to the sinc signal. One problem of window filters is that they are not optimal in the ernel length sense. The frequency of the sinusoidal oscillation is approximately equal to Fc
34 Windowed Sinc FIR filters: Windows Hamming: Blacman: Hanning:
35 Optimal FIR filters: Remez exchange (Pars McClellan) method This design algorithm allows filter designs that are optimal in the ernel length sense. The usage of the method is as simple as passing the filter specification to the algorithm and receiving its filter ernel.
36 FIR filters: Design method Design method Passband Stopband Transition band Matlab Function Equiripple Ripply Ripply Narrow remez (lf) Least Squares Flat Ripply Narrow firls (lf) Max. Flat (LP) Flat Flat Wide maxflat (*) Least Pth-norm Flat Ripply Narrow firlpnorm (*) Constrained Equiripple Flat Ripply Narrow firceqrip (*) Window ** ** ** fir
37 FIR filters: Implementation Direct form or transversal structure. Symmetry (Linear phase) optimization. Fast convolution (overlap add/save).
38 FIR filters: Direct Form x(n) z - x(n+) z - x(n+m-2) z - x(n+m-) h 0 x h x h M- x h M x + y(n) Computational cost: M MAC operations M RAM positions (filter) M RAM positions (signal) Most DSP architectures are optimized for implementing this structure.
39 FIR filters: Transversal Form x(n) x(n+) x(n+m-2) x(n+m-) h 0 x h x h M- x h M z - + z - + z - x + y(n) Computational cost: M MAC operations M RAM positions (filter) M RAM positions (signal) Most DSP architectures are optimized for implementing this structure.
40 FIR filters: Linear phase form x(n) x(n+m-) h 0 x(n+) x(n+half-) x(half) z - z - z - x(n+m-2) x(n+half+) z - z - z x h x h half- x h half x h (n) h (n) M odd half M even n y(n) half n Computational cost: M/2 (even) M/2+ (odd) MAC operations M/2 (even) M/2+ (odd) RAM positions (filter) M RAM positions (signal)
41 FIR filters: Fast convolution FFT M ifft x(n) X y(n) M H(w) M : 2 N multiple Computational cost: 2 FFT (M. log 2 (M) ) M/2 complex multiplications M RAM positions (filter) M (2M for performance) RAM positions (signal) Overlap add/save algorithm.
42 FIR filters: Wordlength effects ADC quantization noise. SNR ADC ADC Bits Coefficients quantization noise. Roundoff quantization noise. Arithmetic overflow noise
43 FIR filters: Roundoff quantization Non linear model Linear model B bits B bits x 2B bits Q B bits e(n) B bits x + B bits 2B bits B bits 2 2 q σ Q 2 (roundoff noise power) q : ADC quantization step (V) Each quantization process (multiplication) degrades system s noise figure.
44 Infinite Impulse Response IIR
45 IIR filters: General characteristics IIR filter output sample depends on previous input samples and previous filter output samples. Implemented using a recursion equation, instead a convolution and don t have linear phase responses. IIR filters have poles and zeros around the Z plane, they are not always stable. Finite duration of nonzero input values / Infinite duration of nonzero output values. Infinite Impulse Response IIR filters achieves better performance than FIR for the same ernel length. Implementation is more complicated, and requires floating point or large word lengths.
46 IIR filters: Non linear phase response As in analog domain, linear phase could be achieved at expense of loosing other desirable features. In other words, IIR filters are much faster than FIR filters but filter stability and phase distortion is a VERY important issue.
47 IIR filters: Linear phase with bidirectional filtering Bidirectional filtering is a way of forcing an IIR filter to have perfect linear phase, at expense of speed since, at least, the computing cost is duplicated.
48 IIR filters: the recursion equation ( z) Y X ( z) ( z) 0 H M + N b ( ) Z a Z ( ) x(n) b 0 z - b x + x a x z - b N -a M x + + x z - z - y(n) y( n) N 0 b M ( ). x( n ) a( ). y( n ) The recursive coefficients are scaled to yield a 0 Each output is a linear combination of past input and output samples
49 IIR filters: Coefficients calculation Pole-zero placement. Impulse invariance. Matched z-transform. Bilinear z-transform.
50 IIR filters: Analog & Digital For HP: And for LP: And for both: (Time decay) (Cutoff frequency) As in the analog domain, simple filters could be implemented with a few coefficients.
51 IIR filters implementation: Direct Form DF is the most direct way of implementing an IIR filter. It uses two buffers (delay lines) and 2(M+) multiplications and sums. More sophisticated forms optimizes memory usage, coefficients precision sensibility and stability. y( n) x(n) z - z - N 0 b M ( ). x( n ) a( ). y( n ) b 0 x + + b x + b N x a x -a M x z - z - y(n)
52 IIR filters implementation: Direct Form 2 (canonic) DF2 Reduces memory usage by two, since the delay line is shared. w( n) y( n) x( n) N 0 b ( ) M a ( ). w( n ). w( n ) x(n) + + -a x w(n) b 0 z - x + b x + y(n) + -a M x z - b N x +
53 IIR filters: SNR in DF and DF2 quantization point quantization Internal nodepoints x(n) b 0 z - b x + x y(n) x(n) w(n) + x + -a z - z - -a b x + x x + b 0 y(n) z - b N x + + -a M x z - + -a M x z - b N x + The canonic section is susceptible to internal self-sustaining overflow. In practice, the poles of the canonic section tend to amplify noise generated during computation more.
54 IIR filters implementation: Design Pitfalls IIR implementation is conditioned mainly by finite wordlength effects. Coefficient quantization, overflow and roundoff errors are most common problems. To go through these problems, some precautions must be taen into account.
55 IIR filters: Overflow The final result overflows. Intermediate results overflow. (Modulo math) Unity gain filters can have outputs that are greater than ± with a bounded input.
56 IIR filters: Noise Noise is generated by truncating high resolution results. The noise generated by truncation often resembles the spectrum of the signal being filtered. The generated noise is most harmful when it recirculates through the recursive coefficients. Force correlated truncation noise to be random (white) by using dither. If the truncation process is deterministic then correction can be applied to the filter. Fixed point truncation is deterministic. Floating point truncation is not deterministic.
57 IIR filters implementation: Coefficient quantization sensibility Frequency response are severely affected for small wordlengths. The sensibility grows with the order of the system. Second order partitioning mitigate this problem by diminishing speed. Instability can occur for very sharp responses.
58 IIR filters implementation: Coefficient quantization sensibility M b b b z b z a z b z H z a z y z b z x z y M q N M N M,,2, ) (, ) ( ) ( ) ( 0 0 K + + N a a a M b b b z a z H q q N q q q,,2,,,2, ) ( 0 K K Π + Π + N i i q q N N q q q q q N N a p p p p p p z p z a z D z D z N z H z p z a z D z D z N z H, ) ( ) (, ) ( ) ( ) ( ) ( ) (, ) ( ) ( ) ( N N i l l l i N i i a p p p p ) (
59 IIR filters implementation: Coefficient quantization sensibility Second order sections (SOS) can be grouped in cascade or parallel. x(n) H H N y(n) When cascading SOS s the order is chosen to maximize SNR. Cascade is the most typically used in DSP processors. x(n) H H N + y(n)
60 IIR Filter: SOS Forms Mar Allie comp.dsp 2004
61 Direct Form I Direct Form I e(n) x(n) N b 0 N 2N N 2N y(n) N (lo) Q y t (n) N(hi) Q y t (n) Z - N N Z - N b 2N 2N -a N Z - N N Z - N b 2 2N 2N -a 2 N Y t b + b z 0 2 ( z) X ( z) E( z) + a z + b + a 2 z z 2 + a z + a z 2 Fixed Point: Modulo math useful. One summer before saturator. No Input scaling required to prevent internal overflow. One truncation feedbac error term. Truncation noise affects recursive terms. Truncation noise negligible for N large enough. Mar Allie comp.dsp 2004 Floating point: No input scaling required. No overflow concerns. Truncation error is coefficient dependant and not correctible. Truncation noise affects recursive terms. Number of bits in mantissa critical.
62 Direct Form II e (n) Direct Form II e 2 (n) x(n) N N 2N N (lo) Q N (hi) w(n) b 0 N 2N y(n) N (lo) Q N (hi) y t (n) N 2N -a Z - w(n-) b N 2N N 2N -a 2 Z - b 2 N 2N w(n-2) Fixed Point: Modulo math not useful. There are 2 sums separated by a saturator. Input scaling required to prevent internal overflow. One truncation feedbac error term. Truncation noise affects all terms. Truncation noise negligible for N large enough. Mar Allie comp.dsp 2004 Floating point: No input scaling required. No overflow concerns. Truncation error is coefficient dependant and not correctible. Truncation noise affects all terms. Number of bits in mantissa critical.
63 Direct Form I Transposed Mar Allie comp.dsp 2004
64 Direct Form II Transposed Direct Form II Transposed e 3 (n) x(n) N b 0 b N 2N N (hi) N 2N Z - Q N 2N y(n) N (lo) e (n) -a N (lo) Q N (hi) y t (n) Q y t (n) Fixed Point: Modulo math useful. One summer before saturator. No Input scaling required to prevent internal overflow. Multiple truncation feedbac error terms*. Truncation noise affects recursive terms. Truncation noise negligible for N large enough. * One if Q and Q2 are removed. Dattorro b 2 2N N N N (hi) 2N Z - Q 2N N N (lo) e 2 (n) -a 2 Floating point: No input scaling required. No overflow concerns. Truncation error is coefficient dependant and not correctible. Truncation noise affects recursive terms. Number of bits in mantissa critical. 2N N 2N N Mar Allie comp.dsp 2004
65 Tabulated Results Fixed Point DFI DFII DFIT DFIIT Floating Point DFI DFII DFIT DFIIT Input scaling Modulo math + + Truncation noise Performs well with 32 bit mantissa Error compensation Good Attribute + Qualified Good Attribute - Poor Attribute Mar Allie comp.dsp 2004
66 IIR Filters: Good News DFI and DFIIT both wor. A processor with 32 bit integer native word size capabilities doesn t need truncation error compensation. Some modern floating point processors can do this inexpensively and fast. Some modern fixed point processors can do this inexpensively and more slowly.
67 IIR Filters: Floating Point All DF IIR filters can wor. Complicated analysis and addend evaluation. Floating point processors with extended precision results may wor well. Analog devices has processors with 32 bit mantissa capabilities. Fixed point processing at 32 bits a sure bet
68 Filters examples
69 DC Blocer Pole/Zero Plot Imaginary Part Real Part
70 DC Blocer: BF53X Fixed Point Q.3 Product Roundoff error! 0 Product and Accumulator error 0 Coefficient quantization error! 0
71 Envelope detection
72 Envelope detection h() h(3) h(5) h(7) h(9) h() h(3) 0 Hilbert Filter
73 Decimation fs, old fs, new, xnew( m) xold ( Mn) M Polyphase implementation F s,new 2B F s,new < 2B
74 Interpolation f s, new L f s, old Polyphase implementation
75 Efficient D/A conversion in compact hi-fi system
76 Equalizing Techniques Flatten DAC Frequency Response
77 Application in the acquisition of high quality data
78 Recommended bibliography EC Ifeachor, BW Jervis. Digital Signal Processing. A Practical approach. Second Edition. Prentice Hall Ch6: A framewor for filter design. Ch7 & 8: FIR & IIR filter design. Ch3: Analysis of wordlength effects in fixed point DSP systems. RG Lyons, Understanding Digital Signal Processing. Third Edition. Prentice Hall 200. Ch5 & 6: FIR & IIR filters. SW Smith, The Scientist and Engineer s guide to DSP. California Tech. Pub Ch4-2: FIR & IIR filter design. J. Dattorro, The Implementation of Recursive Digital Filters for High Fidelity Audio, J. Audio Eng. Soc., vol 36, pp (988 Nov) NOTE: Many images used in this presentation were extracted from the recommended bibliography.
79 Questions? Than you!
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