GCT535- Sound Technology for Multimedia Filters Graduate School of Culture Technology KAIST Juhan Nam 1
Filters § Control the frequency response of input signals – Lowpass, Highpass, Bandpass, Bandreject (notch), Allpass, Equalizers – Shape spectrum of input signals LP BP All pass | H ( f )| | H ( f )| | H ( f )| 0 0 0 f c f /Hz f cl f ch f /Hz f /Hz HP BR | H ( f )| | H ( f )| f b f b + 12 + 6 0 f − 6 − 12 f c f c f c f c 0 0 f c f /Hz f cl f ch f /Hz [DAFx book] 2
Filters in Audio Systems § Tone control – Synthesizers • VCF in Analog synth – Guitar Effect • EQ, Wah Wah Synthesizer (MiniMoog) – DJ mixer, audio Mixers • Filter, EQ – Audio Players • EQ Wah-Wah Pedal 3 Audio Mixer
Filters in Audio Systems x 2 BP1 ( n ) ( ) 2 x RMS1 ( n ) BP 1 LP x 2 BP2 ( n ) § Communication ( ) 2 x RMS2 ( n ) BP 2 LP x ( n ) – Speech Coding • Vocoder: analysis/synthesis of voice x 2 BP k ( n ) ( ) 2 x RMS k ( n ) BP k LP • Formant modeling: linear prediction coding – Audio Coding • Filterbank in MP3 Channel Vocoder – Band-limiting − 20 • Control bandwidth: 8K, 16K • Anti-aliasing filters − 40 − 60 − 80 − 100 0 2 4 6 8 f /kHz → LPC modeling of Formant 4
Filters in Audio Systems § Audio analysis – Short-time Fourier Transform (as a filterbank) – Constant-Q transform – Mel-scaled filterbank – Cochlear filterbank in human ears Oval window Round window Cochlear Filterbank 5
Description of Filter Characteristic § Bode Plot – Log scale in both magnitude and frequency axes Lowpass Filters Lowpass Filters 30 30 Q or Resonance 20 20 Q =4 10 10 Q =2 Gain(dB) Gain(dB) f=400 f=1000 f=3000 f=8000 Q =1 0 0 Passband Q =0.5 − 10 − 10 Roll-Off (e.g. -12dB/oct) Cut-off Frequency − 20 − 20 Stopband − 30 − 30 2 3 4 2 3 4 10 10 10 10 10 10 freqeuncy(log10) freqeuncy(log10) 6
Designing Digital Filters Approach 1 § – Find filter coefficients that satisfy “filter specification” – Low-order filters • Gain at DC and Nyquist, cut-off frequency, Q – High-order filters • Passband/Stopband margin and flatness, width of transition band • Window-based method • Parks-McClellan: Iterative search filter specification • Linear programming or convex optimization • mainly for band-limiting or filterbank – Fitting filters to measured frequency response We handle these two approaches, • Linear prediction coding (LPC) which are mainly used in tone control Approach 2 § 7 – Digitizing analog filters which is already well-developed – Biquad filters: “prototype” analog filters (tone control) – High-order IIR filters: Butterworth filter, Chebyshev filter, Elliptical filter
One-pole One-zero Filters § Three degrees of freedom 𝐵(𝑨) = 𝑐 , + 𝑐 . / 𝑨 0. 𝐼 𝑨 = 𝐶(𝑨) – Fixed: gain at DC (0 Hz) and Nyquist frequency 1 + 𝑏 . / 𝑨 0. • Lowpass: 𝐼 1 = 1 , 𝐼 −1 = 0 • Highpass: 𝐼 1 = 0 , 𝐼 −1 = 1 – User parameter: cut-off frequency -3dB -3dB Cut-off frequency = 1kHz Cut-off frequency = 1kHz Lowpass Filter Highpass Filter 8
One-pole One-zero Filters § Transfer Function 1 + 𝑨 0. 1 − 𝑨 0. 𝐼 𝑨 = 1 − 𝑏 𝐼 𝑨 = 1 + 𝑏 / / 1 − 𝑏 / 𝑨 0. 1 − 𝑏 / 𝑨 0. 2 2 Z-plane Lowpass Filter Highpass Filter 9
One-pole One-zero Filters § Associating the cut-off frequency with the coefficient – Frequency when the gain is -3dB (𝜕 4 1 − tan 2) 1 + 𝑓 067 1 − 𝑏 = 1 4 = 𝑏 = 𝐼 𝜕 / (𝜕 1 − 𝑏 / 𝑓 067 2 2 1 + tan 2) 10
One-pole One-zero Shelving Filter § Equalizers that control bass or treble gain – User parameter: cut-off frequency and band gain à crossover frequency and gain – Fixed: gain at DC or Nyquist • Lowpass: 𝐼 −1 = 0, Highpass: 𝐼 1 = 1 Gain = 12dB Gain = 12dB 9dB 9dB -9dB -9dB Crossover Freq.= 100Hz Crossover Freq. = 1kHz Gain = -12dB Gain = -12dB Bass Shelving Filter Treble Shelving Filter 11
One-pole One-zero Shelving Filter § Transfer Function – Bass Shelving Filter = 10 (ABCD 𝐼 𝑨 = 1 + / 𝑀𝑄 𝑨 4, ) − 1 (Gain ≥ 0dB) 1 @ = 10 (0ABCD 𝐼 𝑨 = ) − 1 1 + @ / 𝑀𝑄 𝑨 (Gain < 0dB) 4, – Treble Shelving Filter 𝐼 𝑨 = 1 + / 𝐼𝑄 𝑨 (Gain ≥ 0dB) 1 𝐼 𝑨 = 1 + @ / 𝐼𝑄 𝑨 (Gain < 0dB) 12
� � Alternative: One-pole One-zero Shelving Filter § Set the gain at crossover frequency to half the maximum gain – Bass: 𝐼 1 = , 𝐼 𝑓 67 K = , 𝐼 −1 = 1 – Treble: 𝐼 1 = 1 , 𝐼 𝑓 67 K = , 𝐼 −1 = Crossover Freq.= 1kHz Crossover Freq.= 1kHz 13
� � � � � � � � Alternative: One-pole One-zero Shelving Filter § Associating the crossover frequency with the coefficients 4 𝑐 , + 𝑐 . 4 = = 4 𝐼 1 𝑏 , − 𝑏 . 𝑐 . = 𝐻 tan 𝜕 , 𝑐 , = 𝐻 tan 𝜕 , − 𝐻 + 𝐻 4 𝑐 , − 𝑐 . Bass 2 4 = 2 𝐼 −1 = 1 𝑏 . = tan 𝜕 , 𝑏 , = tan 𝜕 , 𝑏 , + 𝑏 . − 𝐻 + 𝐻 2 4 2 𝑐 , + 𝑐 . / 𝑓 067 K 4 = 𝐼 𝑓 67 K = 𝑏 , − 𝑏 . / 𝑓 067 K 4 𝑐 , + 𝑐 . 4 = 𝐼 1 = 1 𝑏 , − 𝑏 . tan 𝜕 , tan 𝜕 , 𝑐 . = 𝐻 − 𝐻 𝑐 , = 𝐻 + 𝐻 4 Treble 𝑐 , − 𝑐 . 2 2 4 = = 4 𝐼 −1 tan 𝜕 , tan 𝜕 , 𝑏 , + 𝑏 . 𝑏 . = 𝐻 − 1 𝑏 , = 𝐻 + 1 2 4 2 𝑐 , + 𝑐 . / 𝑓 067 K 4 = 𝐼 𝑓 67 K = 𝑏 , − 𝑏 . / 𝑓 067 K 14
Alternative: One-pole One-zero Shelving Filter The cascade of bass and treble shelving filters renders flat response. 15
Biquad (two-pole two-zero) Filters § Five degrees of freedom – Fixed: gain at DC (0 Hz) and Nyquist frequency – User parameters 𝐵(𝑨) = 𝑐 , + 𝑐 . / 𝑨 0. + 𝑐 4 / 𝑨 04 • Cut-off frequency 𝐼 NO 𝑨 = 𝐶(𝑨) 1 + 𝑏 . / 𝑨 0. +𝑏 4 / 𝑨 04 • Resonance • Bandwidth (sharpness of peak) § Mapping user parameters to filter coefficients – Reson filter can be used but, actually, it is quite complicated – The art of analog filter design is highly advanced and so we take advantage of it 16
Laplace Transform § Laplace Transform: continuous-time version of z-transform X X 𝑌 𝑡 = R 𝑦(𝑢)𝑓 0UV 𝑒𝑢 𝑌 𝑨 = Y 𝑦 𝑜 / 𝑨 0[ , [\, Laplace Transform Z-Transform § Fourier Transform: continuous-time version of discrete-time Fourier Transform X X 𝐼 𝑓 67 = Y ℎ 𝑜 / 𝑓 067[ 𝑌 𝑔 = R 𝑦(𝑢)𝑓 067V 𝑒𝑢 , [\, Fourier Transform Discrete-Time Fourier Transform 𝑡 = 𝑘𝜕 𝑨 = 𝑓 67 17
Example: Laplace Transform § RC lowpass filter – Kirchhoff’s law: 𝑤 a 𝑢 = 𝑤 b 𝑢 + 𝑤 c (𝑢) 𝑒𝑅(𝑢) = 𝑗 𝑢 = 𝐷 𝑒𝑤 c (𝑢) (JOS Filter book) – For capacitor: 𝑒𝑢 𝑒𝑢 𝑒𝑦 𝑗 𝑢 = 𝑤 b (𝑢) – For resistor: 𝑒𝑢 ⇔ 𝑡𝑌 𝑡 − 𝑦(0) 𝑆 c 𝑡 − 𝐷𝑤 0 = 𝑊 b (𝑡) – By taking Laplace transform: 𝐽 𝑡 = 𝐷𝑡𝑊 𝑆 𝐼 𝑡 = 𝑊 c (𝑡) 𝑊 c (𝑡) 1 + 𝑡𝑆𝐷 = 1 1 1 – Transfer function: a (𝑡) = c (𝑡) = 𝑆𝐷 ( ) 𝑡 + 1 𝑊 𝑊 b 𝑡 + 𝑊 𝑆𝐷 (Assuming the initial voltage is zero) 18
Example: Laplace Transform § RLC lowpass filter – Kirchhoff’s law: 𝑤 a 𝑢 = 𝑤 b 𝑢 + 𝑤 c (𝑢) , 𝑤 k 𝑢 = 𝑤 l 𝑢 𝑗 b 𝑢 = 𝑗 l 𝑢 + 𝑗 k (𝑢) (JOS Filter book) 𝑤 l 𝑢 = 𝑀 𝑒𝑗 l (𝑢) – For inductor: ⇔ 𝑊 l 𝑡 = 𝑡𝑀𝐽 l (𝑡) 𝑒𝑢 𝑗 c 𝑢 = 𝐷 𝑒𝑤 c (𝑢) k 𝑡 = 𝐽 l (𝑡) – For capacitor: ⇔ 𝑊 𝑒𝑢 𝑡𝐷 𝑗 b 𝑢 = 𝑤 b (𝑢) – For resistor: ⇔ 𝑊 b 𝑡 = 𝑆𝐽 b (𝑡) 𝑆 𝐼 𝑡 = 𝑊 c (𝑡) 𝑊 c (𝑡) 𝑊 c (𝑡) – Transfer function: a (𝑡) = c (𝑡) = 𝑊 𝑊 b 𝑡 + 𝑊 𝑆 𝑊 l 𝑡 + 𝑡𝐷𝑊 k 𝑡 + 𝑊 c (𝑡) 𝑡𝑀 1 = 1 𝑡 = 1 𝑡 4 + 1 𝑆𝐷 𝑡 + 1 𝑆𝐷 𝑆 𝑡𝑀 + 𝑡𝐷 + 1 𝑀𝐷 19
LTI System in continuous-time domain § In general, Laplace transform of LTI systems is represented with: – Compared to z-transform, 𝑨 0. is replaced with 𝑡 𝑌(𝑡) = 𝐼 𝑡 = 𝑐 , + 𝑐 . / 𝑡 + 𝑐 4 / 𝑡 4 + … + 𝑐 q / 𝑡 q 𝑍(𝑡) 1 + 𝑏 . / 𝑡 + 𝑏 4 / 𝑡 4 + … + 𝑏 r / 𝑡 r § Poles and Zeros – roots of denominator and numerator: 𝑞 O (poles), 𝑟 O (zeros) – Plot the frequency response of the system using poles and zeros 𝐼 𝑡 = 𝑡 − 𝑟 . 𝑡 − 𝑟 4 𝑡 − 𝑟 s … (𝑡 − 𝑟 q ) 𝑡 − 𝑞 . 𝑡 − 𝑞 4 𝑡 − 𝑞 s … (𝑡 − 𝑞 r ) 20
Frequency Response § Frequency response 𝐼 𝜕 can be obtained from 𝐼 𝑡 when 𝑡 = 𝑘𝜕 – Analogous to frequency response 𝐼 𝜕 when 𝑨 = 𝑓 67 at z-transform s-plane z-plane 𝑨 = 𝑓 67 𝑡 = 𝑘𝜕
� Example: RC lowpass filter § Transfer function 𝑏 (𝑏 = 1 𝐼 𝑡 = 𝑆𝐷) 𝑡 + 𝑏 § Amplitude response ≈ v – for 𝜕 ≪ 𝑏 , 𝐼 𝑘𝜕 v = 1 = 0𝑒𝐶 v v . 4 – at 𝜕 = 𝑏 , 𝐼 𝑘𝜕 = 67wv = 6vwv = 6.w. = 4 = −3𝑒𝐶 67 = v v – for 𝜕 ≫ 𝑏 , 𝐼 𝑘𝜕 ≈ 7 § Phase response – for 𝜕 ≪ 𝑏 , ∠𝐼 𝑘𝜕 = 0 – at 𝜕 = 𝑏 , ∠𝐼 𝑘𝜕 = −45° – for 𝜕 ≫ 𝑏 , ∠𝐼 𝑘𝜕 = −90° 22
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