Peaking and Shelving Filter Properties

Transcription

1 MUS424: Signal Processing Techniques for Digital Audio Effects Handout #22 Jonathan Abel, David Berners May 18, 24 Lecture #14: May 18, 24 Lecture Notes 12 Peaking and Shelving Filter Properties Shelf Filter Specification A shelf filter h(ω) takes on a gain of l at DC, a gain of l π at high frequencies, and a gain of (l l π ) 1 2 at a transition frequency ϕ. A low shelf filter has the high-frequency gain l π fixed at one, with l free to vary. In a high shelf filter, the high-frequency gain l π is varied with the DC gain l fixed at one.

2 2 MUS424: Handout #22 First-Order Shelf Filter Analog Prototype Consider the following first-order analog prototype filter, h(s) = l π s/ρ + l. s/ρ + 1 Note that the analog prototype filter takes on the desired DC and high-frequency gains, h(ω = ) = l, h(ω ) = l π, with the factor ρ controlling the transition frequency. The analog prototype filter is designed to have a transition frequency of one. Setting h(j 1) 2 = l l π, the factor ρ may be specified. h(j 1) 2 = l l π = l2 π/ρ 2 + l 2 1/ρ 2 + 1, which implies ρ = ( lπ l )1 2. Substituting into the expression for h(s), we have h(s) = (l l π ) 1 2 s + (l /l π ) 1 2 (l /l π ) 1 2s + 1.

3 MUS424: Handout #22 3 First-Order Shelf Filter Analog Prototype Properties The analog prototype shelf filter has transfer function h(s) = (l l π ) 1 2 s + (l /l π ) 1 2 (l /l π ) 1 2s + 1. It has a real pole and real zero at reciprical frequencies about the transition frequency s = j 1, s p = (l π /l ) 1 2, s z = (l /l π ) 1 2. Note that on a db scale the normalized analog shelf filter is antisymmetric in log s about the transition frequency, h(1/s) 1/s + (l /l π ) 1 2 (l /l π ) 1 2/s + 1 = 1 + (l /l π ) 1 2s (l /l π ) s 1/h(s).

4 4 MUS424: Handout #22 First-Order Digital Shelf Filter via Bilinear Transform The digital shelf filter is formed via bilinear transform on the analog prototype, We have and, h(z) = h(s(z)), s = 2 T 1 z z 1. h(s) = (l l π ) 1 2 s + 1/ρ s/ρ + 1, ρ = (l π/l ) 1 2, h(z) = (l l π ) 1 2 ρ(1 z 1 ) + T 2 (1 + z 1 ) (1 z 1 ) + ρ T 2 (1 + z 1 ), = (l l π ) 1 ( T ρ) + (T 2 ρ)z 1 (ρ T 2 + 1) + (ρt 2 1)z 1, [ ] ( ) = (l l π ) 1 ρ + T 1 1 T/2ρ z T/2ρ 1 + ρ T. 2 1 )z 1 ( 1 ρt/2 1+ρT/2 Note that the pole at ρ and zero at 1/ρ have been transformed according to the bilinear transform.

5 MUS424: Handout #22 5 Selecting T The sampling period T is chosen to put the transition frequency of the analog prototype, Ω = 1, at the desired frequency ϕ. Evaluating the bilinear transform on the unit circle, z 1 = e jω, s = 2 T 1 z z 1 = 2 T 1 e jω 1 + e jω, = 2 T ejω/2 e jω/2 e jω/2 + e jω/2 = 2 T j sin(ω/2) cos(ω/2), we see that the frequency ω on the unit circle corresponds to the frequency Ω = 2 T tan(ω/2) on the s-plane imaginary axis. (Note that for small frequencies ω 1, tan(ω/2) ω/2, and Ω 1 T. To map the analog prototype filter transition frequency of 1 to the desired transition frequency ϕ, we set T 2 = tan(ϕ/2).

6 6 MUS424: Handout #22 Example Shelf Filters 2 second-order shelf filter transfer function magnitudes, ft = 125 * 2.^[:6] second-order shelf filter transfer function magnitudes, gain = [2:2:2]

7 MUS424: Handout #22 7 Peak Filter Specification A peak (or notch) filter p(ω) takes on a gain of 1 at DC and high frequencies, and achieves a maximum (or minimum) gain of l at some point between the transition frequencies ϕ ±, at which the gain is l. The frequency of the magnitude extremum is called the center frequency, ϕ c.

8 8 MUS424: Handout #22 Digital Peak Filter The second-order digital filter with coefficients given by p(z) = b + b 1 z 1 + b 2 z a 1 z 1 + a 2 z 2 a 2 = 2Q sin ϕ c, 2Q + sinϕ c a 1 = b 1 = (1 + a 2 ) cos ϕ c, b = 1 2 (1 + a 2) (1 a 2)l, b 2 = 1 2 (1 + a 2) 1 2 (1 a 2)l, implements a peak (or notch) filter with maximum (or minimum) gain l at a center frequency ϕ c between the specified transition frequencies ϕ ±, at which the filter takes on magnitude l. The center frequency ϕ c and the inverse bandwidth Q may be written in terms of the transition frequencies ϕ ± and db peak gain λ, { = acos κ sign{κ} ( } κ 2 1 )1 2, ϕ c κ = 1 + cos ϕ cos ϕ + cos ϕ + cosϕ + Q = 1 [ l sin 2 ϕ c (cos ϕ + cos ϕ + ) 2 2 cos ϕ c cosϕ cos ϕ + In the case that ϕ + + ϕ = π, we have ] 1 2. ϕ c = π/2, l Q = 2 cotδ, δ = 1 2 (ϕ ϕ + ).

9 MUS424: Handout #22 9 Example Peak Filters 2 peak filter transfer function magnitudes, ft = 125 * 2.^[:6] peak filter transfer function magnitudes, gain = [2:2:2]

10 1 MUS424: Handout #22 Graphic Equalizer 6 graphic equalizer shelf and peak filters A graphic equalizer is implemented as a cascade of peak and shelf filters having a prioi specified bandwidths and user-controlled gains.

11 MUS424: Handout #22 11 Graphic Equalizer Behavior 1 traditional graphic equalizer with controls set to [ 5 5 ] db By adjusting the gains of the peak and shelf filters in the cascade, a wide variety of transfer function magnitudes may be produced. Although the idea is that the transfer function magnitude should smoothly interpolate the specified gains, it doesn t always work out that way.

12 12 MUS424: Handout #22 Graphic Equalizer Behavior 1 traditional graphic equalizer with controls set to 5*ones(1,8) db Contributions from adjacent bands cause the filter to overshoot the desired gains.

13 MUS424: Handout #22 13 Graphic Equalizer Behavior 1 traditional graphic equalizer with narrow-band peak filters Traditional graphic equalizers don t smoothly interpolate the given gains. Small filter bandwidths lead to rippled transfer functions. Large filter bandwidths overshoot desired gains.

14 14 MUS424: Handout #22 Alternative Gain Computation 1 novel graphic equalizer The idea is to find a set of peak and shelf filter gains which account for the overlap between bands, so that the resulting peak and shelf filter cascade interpolates the specified band gains.

15 MUS424: Handout #22 15 Peak Filter Self Similarity 2 peak filter transfer function magnitudes, gain = [2:2:2] normalized peak filter transfer function magnitudes, gain = [2:2:2] Peaking filters parameterized by a maximum db gain λ, achieved somewhere between transition frequencies ϕ and ϕ +, at which the db gain is λ/2, are approximately self similar on a log magnitude scale. α log p(ω; λ,ϕ ± ) log p(ω;α λ,ϕ ± ).

16 16 MUS424: Handout #22 Shelf Filter Self Similarity 2 second-order shelf filter transfer function magnitudes, gain = [2:2:2] normalized shelf filter transfer function magnitudes, gain = [2:2:2] Low shelf filters specified by a low-frequency db gain λ, a db gain λ/2 at a transition frequency ϕ, and a high-frequency gain of one are also approximately self similar on a log magnitude scale. α log h(ω;λ, ϕ) log h(ω;α λ,ϕ).

17 MUS424: Handout #22 17 Filter Design Approach Consider a cascade g(ω; θ) of K peak and shelf filters having gains and λ k, k = 1,...,K and transition frequencies ϕ k, k = 1,...,K 1 stacked in the column θ, g(ω; θ) = h(ω; λ 1,ϕ 1 ) h(ω;λ K,ϕ K 1 ) K 1 p(ω; λ k,ϕ k 1, ϕ k ). k=2 Because of the self similarity property, the db magnitude of the cascade, denoted by γ(ω;θ), γ(ω;θ) = def 2log 1 {g(ω;θ)}, = σ(ω; λ 1,ϕ 1 ) + σ(ω; λ K,ϕ K 1 ) + K 1 π(ω;λ k,ϕ k 1,ϕ k ), k=2 is approximately linear in the filter gains, γ(ω; θ) λ 1 σ(ω; 1.,ϕ 1 ) + λ K σ(ω; 1.,ϕ K 1 ) + K 1 λ k π(ω; 1., ϕ k 1,ϕ k ). k=2

18 18 MUS424: Handout #22 Filter Design Approach At a particular frequency ω i, we have γ(ω i ; θ) [ σ 1 (ω i ) π 2 (ω i ) π K 1 (ω i ) σ K (ω i ) ] λ, where λ is the stack of db band gains, λ = [ λ 1 λ K ], and where σ k (ω i ) and π(ω i ) are the transfer function db magnitudes of shelf and peak filters with specified transition frequencies and 1. db nominal gains, evaluated at ω i, and σ k (ω i ) = 2 log 1 {h(ω i ; 1.dB,ϕ k )} π(ω i ) = 2 log 1 {p(ω i ; 1.dB,ϕ k 1,ϕ k )}. Stacking instances of γ(ω; θ) evaluated at a set of frequencies ω i to form the column γ, we have γ Bλ, B = [ σ 1 π 2 π K 1 σ K ], where σ k and π k are columns of db magnitudes evaluated at ω i of shelf and peak filters having gains of 1. db and specified transition frequencies.

19 MUS424: Handout #22 19 Graphic Equalizer Design The self similarity of second-order peaking and shelving filters leads to an approximate linear relationship between the db filter gains and the db gain of the cascade, γ Bλ, Therefore, given a set of shelf and peak filters having specified transition frequencies, and positive definite weighting matrix W, the gains ˆλ = (B WB) 1 B Wγ will approximately minimize the weighted square difference between a desired db magnitude response γ and the shelf and peak filter cascade db magnitude at frequencies ω i, γ. For a graphic equalizer with K 1 fixed band edges, the frequencies ω i can be chosen as the K band centers, and the gains ˆλ simply computed as the control gains γ scaled by the basis inverse, ˆλ = B 1 γ.

20 2 MUS424: Handout #22 Graphic Equalizer Design Example traditional and novel graphic equalizers For a graphic equalizer with K 1 fixed band edges, the frequencies ω i can be chosen as the K band centers, and the gains ˆλ simply computed as the control gains γ scaled by the basis inverse, ˆλ = B 1 γ. Note that for a grphic equalizer with fixed band edges, B 1 may be computed a priori. To account for discrepancies in the self similarity property, ˆλ may be computed iteratively, forming B using the gains from the previous solution.

21 MUS424: Handout #22 21 Iterated Solution graphic equalizer with iterated band gains Iterative solution of the equation for the corrected band gains can correct for discrepancies in filter self similarity, λ n+1 = B(λ n ) 1 γ. It seems likely that the iteration converges since the sensitivity of the basis to changes in the gain are small, and changes in the basis B are monotonic with changes in the gains λ.

22 22 MUS424: Handout #22 Transfer Function Modeling measured transfer function magnitude, selected extrema A cascade of peak and shelf filters can be fit to an arbitrary transfer function by selecting a set of transition frequencies, and fitting the gains. Transition frequency selection method. Tabulate extrema or inflection point frequencies. Pick transition frequencies as geometric means of significant extrema or at significant inflection points.

23 MUS424: Handout #22 23 Transfer Function Modeling measured (solid) and modeled (dashed) transfer function magnitudes Band gain selection method. Pick a dense sampling of frequencies ω i, say Bark or ERB spaced. Form basis, and compute gains, ˆλ = (B WB) 1 B Wγ. The resulting transfer function will not interpolate the specified gains γ, but rather approximate the desired magnitude, minimizing the weighted mean square db difference.

24 24 MUS424: Handout #22 Critical Bandwidth and Critical Band Smoothing 1 original and critical-band smoothed power spectra When applied to broad band signals, details of a filter smaller than a critical bandwidth are irrelevant only the total power matters.

25 MUS424: Handout #22 25 Critical Bandwidth The Bark frequency scale divides the human hearing range into about 25 ciritcal bands. Frequency f in khz and critical-band rate b in Bark are related via the following expressions, b = 1.3 [ log(1 + f 2 ) ] 1 2, f = [ exp { (b/1.3) 2} 1 ] 1 2.

26 26 MUS424: Handout #22 Critical Bandwidth Critical bandwidth as a function of frequency is approximately w = f 3 2.

27 MUS424: Handout #22 27 Critical Band Smoothing 3 original and smoothed power spectra By smoothing the filter power over bandwidths proportional to critical bandwidth, filter complexity is reduced while retaining psychoacoustically relevant cues. Critical-band smoothing may be done via a running mean over frequency, or via windowing of a frequencywarped impulse response.

28 28 MUS424: Handout #22 Critical Band Smoothing 7 original and smoothed impulse responses time - milliseconds As expected, impulse responses are shortened by criticalband smoothing.

29 MUS424: Handout #22 29 Filter Phase 3 raw, minimum phase and linear phase impulse responses time - milliseconds Humans are relatively insensitive to filter phase. A linear phase filter delays all frequency components the same amount τ, H(ω) = e jτω H(ω). A minimum phase filter delays input energy the least of all filters having the same magnitude response, m m h 2 min (n) h 2 (n), m. n= n=