Digital Signal Processing-Digital Filters
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This document discusses digital signal processing using digital filters in MATLAB. It begins by introducing signals and their analog and digital processing. It then covers key digital signal processing tasks like filtering, transforms, and convolution. It describes different filter types including FIR and IIR, and filter design methods. MATLAB sessions are included to demonstrate filtering and filter design. The overall document provides a conceptual overview of digital filters and digital signal processing.
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Digital Signal Processing-Digital Filters
- 1. DSPing with Digital Filters An simulation through MATLAB advanced tools This work is licensed under a Creative Commons Attribution 2.5 India License. By nmx prime @ssh daemon catch~me<at>nelsonanand@gmail.com
- 2. Signals • Exists as – Physical quantity such as force, velocity, vibration – Electrical such as electric & magnetic fields – Optical quantity, the Photon – Energy, the Quantum • These are continuous i.e., analog in nature
- 3. Analog Processing • Analog is continuously changing w.r.t time • Before 19th century, we don’t have digital world, where analog way is only available signal processing tool • Analog processing includes filtering, impedance matching, transistor based designs, etc.,
- 4. Digital signal processing • When signal is processed as integers & floating numbers, it’s DSP. • Requires converting physical quantity to numerical values, accompanied through ADC • Even, we require less complication; so, we digitized the Discrete domain, where all dimensions are discrete
- 5. DSP Tasks,(usual) • Filtering – Removing unwanted frequencies from a signal. • Spectrum Analysis – Determining the features constellated in frequencies of a signal. • Synthesis – Generating complex signals such as speech. • System Identification – Identifying the properties of a system by numerical analysis. • Compression – Reducing the memory or bandwidth it takes to store a signal, such as audio or video.
- 6. Convolution • Linear • Circular Transforms • DFT • DCT • DST • Wavelets & others
- 7. Convolution • Mathematical way of combining two signals to form a third signal. • Resolving of a continuous function. • Otherwise called , helps to analyze any complex signal as one sample per instant. • Indeed it’s sampling &actually filtering. • This results in Impulse Response of LTI systems Tells why shd use impulse response & why every sys has to be LTI
- 8. Convolution (cntd.,) • Types –based on the graphical orientation, –Linear Resolute a continuous line into segments –Circular Resolute a circle into wheel of equal separation
- 9. Linear Convolution • Allows sequences of unequal lenghts to be convolved. • Let 2 sequences of length M and N (M≠N). The resulting convolved signal will be of length M+N-1.
- 10. Linear Convolution • Allows sequences of unequal lenghts to be convolved. • Let 2 sequences of length M and N (M≠N). The resulting convolved signal will be of length M+N-1.
- 11. Convolution do filtering Img courtesy: White MS, Delmar EE series
- 12. Fourier approach • Split & work efficiently.. – Split complex signal into easier sinusoids… • In the jargon of signal processing, the input and output signals are viewed as a superposition (sum) of simpler waveforms. • This is the basis of nearly all signal processing techniques.
- 13. • Jean Baptiste Joseph Fourier (1768-1830), • 4 types –Aperiodic-Continuous Fourier Transform –Periodic-Continuous Fourier Series –Aperiodic-Discrete Discrete Time Fourier Transform. • Periodic-Discrete Discrete Fourier Transform(DFT)
- 14. • acc to review of Joseph Louis Lagrange & Simon de Laplace, Fourier analysis fits to Infinite length sequence • But DSPs process only finite length sequence • Way is “imagined infinitive” – Seeing finite as periodic repetitions & considering single periodic component, the actual signal . • Indeed entire DSP is agreement & understanding ha ha ha!! !! !!
- 15. • Extracting feature from the signal Spectral Analysis of Signals • Describing a system & it’s performance Frequency Response of Systems • Convolution in T.D. is Multiplication of their Fourier Spectrum Convolution
- 16. Spectral Analysis 13 Hz peak due to 3-blade propeller running @4.3 rpm Analysis of submarine movements (SONAR) Img courtesy of Steven W Smith, California Technical Pubs
- 17. Frequency Response of System Img courtesy of Steven W Smith, California Technical Pubs
- 18. Impulse Response (vs) Frequency response Img courtesy of Steven W Smith, California Technical Pubs
- 19. Time Domain (vs) Frequency Domain • TD Natural domain of every timely described signals • FD Analysis & Synthesis Domain • TD compresses FD expands • When the time domain is compressed until it becomes an impulse, the frequency domain is expanded until it becomes a constant value.
- 20. The MAG &PHASE(polar form) Img courtesy of Steven W Smith,California Technical Pubs
- 21. The MAG &PHASE(rectangular form) Img courtesy of Steven W Smith,California Technical Pubs
- 22. Polar Mag & phase Graphs of polar co-ordinate Suitable for representation Rectangular Real & Imaginary Cosine & sine components Suitable for software computations
- 23. sinc & pulse Img courtesy of Steven W Smith, California Technical Pubs
- 24. Convolution from DFT • This is called circular convolution – Multiplication of DFTs of the sequences • The convolving sequences should be of equal length. • If not equal, should be made equal using zero padding[inserting zeros] (agreement of trigonometry)
- 25. DFT Sampled Fourier Tranform. Spectrum viewer, relates the contribution of each frequency to the information
- 26. So ,what’s ω,Ω,F and f F-CTS Frequency (cycles / second) f-DTS Frequency (cycles / sample) t=nT, T is sampling period (seconds/sample) 1/F=n/Fs , Fs is sampling frequency 1/n=F/Fs f=F/Fs Cycles/second Samples/second Cycles Samples Ω=2πF ω=2πf Measured in redians
- 27. convolution transforms Img courtesy of Robert Oshana,Newnes Publications,
- 28. Filtering • separation of signals that have been combined • restoration of signals that have been distorted • every linear filter has an impulse response, a step response and a frequency response.
- 29. Filtering (cntd.,) Img courtesy of Steven W Smith, California Technical Pubs
- 30. Filtering (cntd.,) • impulse response- specifies the filter performance in time domain • step response- describes the waveshape preserving quality of filter – Important when information is coded in the waveshape of the signal(modulations) • frequency response- – Linear scale-passband ripple &role-off(described well on linear) – Logarithmic scale-stopband attenuation(described well on log)
- 31. Time domain parameters • These can be analyzed through step response • Rise & Fall time • Overshoot • Linear phase
- 32. Rise & Fall Time Should be as fast as possible Img courtesy of Steven W Smith, California Technical Pubs
- 33. Overshoot Img courtesy of Steven W Smith, California Technical Pubs distortion of the information contained in the time domain.
- 34. Linear Phase The Symmetry, needed to make the rising edges look the same as the falling edges Img courtesy of Steven W Smith, California Technical Pubs
- 35. Frequency domain parameters Img courtesy of Proakis.G John, Prentice-Hall internationals Passband Role-off
- 36. Frequency domain parameters (cntd.,) • All filters introduce a delay in response, • Measured as group delay & phase delay – Rate of change of phase to the frequency
- 37. Digital Filtering • Filtering through lump of numerical values those are actually real physical quantities • They are called ARMA filters – Auto Regressive Moving Average Filter • Indeed, it’s basically averaging
- 38. To design a filter, we need • Filter specifications, the need • Filter response, how we attain the need • It’s efficiency in terms of Real-Time implementation • This is why algorithms are still developed The DSP System IIR FIR LMS
- 39. Convolution do filtering (duplicated) Img courtesy: White MS, Delmar EE series FIR
- 40. Analysis • Mag res • Phase res Design • FIR/IIR/LMS & others Implement • structure • platform Transfer function Real-time constraints DSP SYSTEM
- 41. Analysis • Mag res • Phase res Design • FIR/IIR/LMS & others Implement • structure • platform Transfer function Real-time constraints DSP SYSTEM
- 42. Response • Butterworth • Bessel • Chebyshev • Inverse chebyshev • Cauer All-pole filters(no zeros)
- 43. Custom responses • Comb • Notch • All pass (sometime “all stop”) • Interpolator • Decimator Multirate Signal Processing
- 44. Butterworth Response No ripple in passband & stopband Long transition region , i.e., skirt length Predictable phase & group delay Src & Img courtesy: White MS, Delmar EE series
- 45. Chebyshev response ripple in passband & no ripples in stopband Shorter skirt length i.e., steep cut-off Approximately predictable phase & group delay Low pass filter Src & Img courtesy: White MS, Delmar EE series
- 46. Inverse chebyshev Ripple in stopband & no ripple in passband Shorter skirt length i.e., steep cut-off Unpredictable phase & group delay Band pass filter Src & Img courtesy: White MS, Delmar EE series
- 47. Cauer response Ripple in pass & stp band Highly distorted phase response Very narrow skirt, i.e., steeper response than other Only predicted data table to design filters Band stop filter Src & Img courtesy: White MS, Delmar EE series
- 48. Analysis • Mag res • Phase res Design • FIR/IIR/LMS & others Implement • structure • platform Transfer function Real-time constraints DSP SYSTEM
- 49. Digital Filters • IIR Infinite Impulse response – Produces infinite output points when excited by an impulse signal(very short signal of area=1) • FIR Finite Impulse response – Produces finite output points of length(size) N when excited by an impulse signal
- 50. IIR Filters • They are sampled versions of analog filters • General DSP equation of IIR filter is Filter co-efficients Previous inputs Previous outputs Filtered output
- 51. IIR Filters • The transfer function model of IIR filters is Zeros poles Transfer function
- 52. Poles & Zeros • Pole location – defines the frequencies to pass • Zero location – defines the frequencies to be stopped
- 53. Poles & Zeros(Contd.,) 0 rad π rad
- 54. IIR filter design • They have poles & zeros as like analog filters tf • So, they have parental s-plane poles & zeros • Hence, analog parameters in s-plane can be transformed into Digital parameters in z-plane
- 56. FIR Filters • No poles no parental s-plane • General DSP equation of FIR filter is Filter co-efficients Previous inputs Filtered output
- 57. Constraint on FIR filter • In DSP jargon, the impulse response is not fully immersed in input signal • This causes a transient in TD • That’s why brick wall filter with transiently spread impulse response is yet possible not in real-time 0 Img courtesy of Steve Winder , Newnes Pubs
- 58. The solution is • Shift negative time to zero time • Approximate the filter kernel Img courtesy of Steven W Smith, California Technical Pubs
- 59. Ensues • Ripples • -(N/2)th sample shifted as 0th sample • 0th sample shifted to (N/2)-1th sample • Or it could be • -1th sample shifted as (N/2)th sample 0-N/2 (N/2)-1 Signal component 0 N-1 Usually ignored
- 60. FIR Types Zero Phase Linear Phase Minimum phase Maximum phase
- 61. FIR Types – Consider M is filter order(taps)-no. of co-eff • Symmetric h(n)=h(M-1-n) & – M is Odd – M is Even • AntiSymmetric h(n)=-h(M-1-n) & – M is Odd – M is Even
- 62. FIR Types Img courtesy of Dan Ellis, ELEN 4810
- 63. FIR Design methods • Frequency Sampling Method • Windowing Technique • Remez Algorithm (aka., equiripple) – Based on Alternation theorem – Implemented by Parks-McClellan Iteration Program
- 65. Various windows on ground Img courtesy of Dan Ellis, ELEN 4810
- 66. A PULL STOP IN A BOOK IS NOT AN END OF UR INTUITION Free Advise Feedbacks are heartily welcomed, Mail to nelsonanand@gmail.com