of Impulse Responses on a Circle. Using a Uniformly Moving Microphone. Nara Hahn and Sascha Spors. Research Group Signal Processing and Virtual ...
Continuous Measurement of Impulse Responses on a Circle Using a Uniformly Moving Microphone Nara Hahn and Sascha Spors Research Group Signal Processing and Virtual Acoustics University of Rostock, Germany
23rd EUSIPCO Nice, France, 4. September 2015
Impulse Response Measuring at Multiple Positions ,k h (x
1
simultaneous (microphone array)
2
sequential (single microphone)
3
continuous (single microphone)
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Continuous IR Measurement
x
| Introduction
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Impulse Response Measuring at Multiple Positions
1
simultaneous (microphone array) ⊕ time-efficient expensive sensor calibration required disturbed sound field
2
sequential (single microphone)
3
continuous (single microphone)
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Continuous IR Measurement
| Introduction
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Impulse Response Measuring at Multiple Positions
1 2
3
simultaneous (microphone array) sequential (single microphone) ⊕ acoustically transparent time-consuming suffers from (long-term) time-variance, e.g. temperature drift and voice-coil heating continuous (single microphone)
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Continuous IR Measurement
| Introduction
2 / 23
Impulse Response Measuring at Multiple Positions
1
simultaneous (microphone array)
2
sequential (single microphone)
3
continuous (single microphone) ⊕ acoustically transparent ⊕ time-efficient consideration of (short-term) time-variance required
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Continuous IR Measurement
| Introduction
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Continuous Measurement , h(n
k)
y (n)
x(n)
fixed loudspeaker + moving microphone periodic excitation identification of linear time-invariant systems by using a linear time-varying system
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Continuous IR Measurement
| Introduction
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Continuous Measurement , h(n
k)
y (n)
x(n)
existing methods [Hulsebos, 2004] [Fukudome, 2007] [Ajdler, 2007] [Antweiler, 2012] [Hahn, 2014]
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Excitation periodic sine sweep maximum length sequence sinusoids perfect sequence perfect sequence
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Continuous IR Measurement
Approach demodulation cross-correlation + smoothing spectral division in the spatial frequency domain adaptive filtering (normalized LMS) estimating orthogonal expansion coefficients
| Introduction
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Linear Time-varying System
finite impulse response (FIR) model
y (n) =
N−1 X
h(n, k)x(n − k)
k=0
k x(n)
input signal
y (n)
output signal
h(n, k)
k -th coefficient at n-th sample
N
length of the impulse response
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h(n0 , k)
Continuous IR Measurement
n0
| System Identification
n
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Perfect Sequence Periodic signal with perfect autocorrelation periodic ψ(n + N) = ψ(n) perfect autocorrelation N−1 X
ψ(n − k)ψ(n − l) =
( 1
n=0
0
(k − l) mod N = 0 otherwise
e.g. maximum length sequence (MLS), periodic sweep, etc. Orthogonal expansion time-shifted sequences form an orthogonal basis set (RN )
n o Ψ = ψ(−k), ψ(1 − k), ψ(2 − k), . . . , ψ(N − 1 − k) orthogonal expansion of impulse responses
h(n, k) =
N−1 X
am (n)ψ(m − k),
k = 0, 1, ..., N − 1
m=0 N.Hahn and S.Spors
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Continuous IR Measurement
| System Identification
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Perfect Sequence Periodic signal with perfect autocorrelation periodic ψ(n + N) = ψ(n) perfect autocorrelation N−1 X
ψ(n − k)ψ(n − l) =
( 1
n=0
0
(k − l) mod N = 0 otherwise
e.g. maximum length sequence (MLS), periodic sweep, etc. Orthogonal expansion time-shifted sequences form an orthogonal basis set (RN )
n o Ψ = ψ(−k), ψ(1 − k), ψ(2 − k), . . . , ψ(N − 1 − k) orthogonal expansion of impulse responses
h(n, k) =
N−1 X
am (n)ψ(m − k),
k = 0, 1, ..., N − 1
m=0 N.Hahn and S.Spors
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Continuous IR Measurement
| System Identification
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Excitation with a Perfect Sequence N−1 X
y (n) =
h(n, k)ψ(n − k)
k=0 ( N−1 X N−1 X
=
k=0 N−1 X
=
m=0
) am (n)ψ(m − k) ψ(n − k)
m=0
am (n)
N−1 X
ψ(m − k)ψ(n − k)
k=0
|
{z
=δm,n mod N
}
= an mod N (n) sequential excitation of the individual orthogonal components
y (n) corresponds to the respective expansion coefficient1 each expansion coefficient is decimated by a factor of N 1
Carini, Efficient NLMS and RLS Algorithms for Perfect Periodic Sequences, ICASSP, 2010.
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Continuous IR Measurement
| System Identification
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Excitation with a Perfect Sequence N−1 X
y (n) =
h(n, k)ψ(n − k)
k=0 ( N−1 X N−1 X
=
k=0 N−1 X
=
m=0
) am (n)ψ(m − k) ψ(n − k)
m=0
am (n)
N−1 X
ψ(m − k)ψ(n − k)
k=0
|
{z
=δm,n mod N
}
= an mod N (n) sequential excitation of the individual orthogonal components
y (n) corresponds to the respective expansion coefficient1 each expansion coefficient is decimated by a factor of N 1
Carini, Efficient NLMS and RLS Algorithms for Perfect Periodic Sequences, ICASSP, 2010.
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Continuous IR Measurement
| System Identification
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y (n) = an mod N (n),
N=4
y (n) n a0 (n) 0
4
8
n
12
a1 (n) 1
5
9
n
13
a2 (n) 2
6
10
14
n
a3 (n) 3
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7
11
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| System Identification
n 7 / 23
y (n) = an mod N (n),
N=4
y (n) n a0 (n) 0
4
8
n
12
a1 (n) 1
5
9
n
13
a2 (n) 2
6
10
14
n
a3 (n) 3
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7
11
Continuous IR Measurement
| System Identification
n 7 / 23
y (n) = an mod N (n),
N=4
y (n) n a0 (n) 0
4
8
n
12
a1 (n) 1
5
9
n
13
a2 (n) 2
6
10
14
n
a3 (n) 3
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Continuous IR Measurement
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n 7 / 23
y (n) = an mod N (n),
N=4
y (n) n a0 (n) 0
4
8
n
12
a1 (n) 1
5
9
n
13
a2 (n) 2
6
10
14
n
a3 (n) 3
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11
Continuous IR Measurement
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y (n) = an mod N (n),
N=4
y (n) n a0 (n) 0
4
8
n
12
a1 (n) 1
5
9
n
13
a2 (n) 2
6
10
14
n
a3 (n) 3
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Continuous IR Measurement
| System Identification
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System Identification 2
interpolate am (n) impulse responses computed as
ˆ k) = h(n,
N−1 X
aˆm (n)ψ(k − m)
m=0
am (n) can be recovered, only if its original bandwidth is less than
fs N
lower-order interpolations seem to be sufficient
2
Nara Hahn and Sascha Spors, Identification of Dynamic Acoustic Systems by Orthogonal Expansion of Time-variant Impulse Responses, ISCCSP 2014. N.Hahn and S.Spors
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Continuous IR Measurement
| System Identification
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Continuous Measurement on a Circle
Ωφ n=0
r0
r0
radius of the circular trajectory
Ωφ
angular speed of the microphone captured sample
length of y (n) effective number of measurements anti-aliasing condition maximum allowable angular speed
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2π Ωφ f s Meff = NL 0 Meff ≥ 2ωr c Ωφ ≤ r0cN
L=
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Continuous IR Measurement
| System Identification
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Continuous Measurement on a Circle
a0 (n)
a1 (n)
length of y (n) anti-aliasing condition maximum allowable angular speed
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4. Sep. 2015
a3 (n)
2π Ωφ f s Meff = NL 0 Meff ≥ 2ωr c Ωφ ≤ r0cN
L=
effective number of measurements
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a2 (n)
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Continuous IR Measurement
| System Identification
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Continuous Measurement on a Circle
a0 (n)
a1 (n)
length of y (n) anti-aliasing condition maximum allowable angular speed
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a3 (n)
2π Ωφ f s Meff = NL 0 Meff ≥ 2ωr c Ωφ ≤ r0cN
L=
effective number of measurements
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a2 (n)
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Continuous IR Measurement
| System Identification
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Measurement Setup Neumann KH120 loudspeaker + omni-directional microphone VariSphear (motorized arm)3 continuous (hc (φ, k)) and sequential (hs (φ, k)) measurements performed
ψ(n) periodic sequence fs
= 44.1 kHz
N
= 88200 (2 s)
r0
= 0.5 m
Ωφ
= 0.25 ◦ /s
Meff
= 720
L
= 1440 × 88200 (25 min)
3
Benjamin Bernschütz et al., Entwurf und Aufbau eines variablen sphärischen Mikrofonarrays für Forschungsanwendungen in Raumakustik und virtual Audio, DAGA, 2010. N.Hahn and S.Spors
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Continuous IR Measurement
| Measurements
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|hc (φ, t)| Continuous measurements (0–22 ms) 22 −30 20 −35 18 −40 16 −45 14 t / ms
−50 12 −55 10 −60 8 −65 6 −70 4 −75 2 −80 0 −150 −100 −50
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0 50 φ / deg
Continuous IR Measurement
100
| Measurements
150
dB
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|hs (φ, t)| Sequential measurements (0–22 ms) 22 −30 20 −35 18 −40 16 −45 14 t / ms
−50 12 −55 10 −60 8 −65 6 −70 4 −75 2 −80 0 −150 −100 −50
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0 50 φ / deg
Continuous IR Measurement
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| Measurements
150
dB
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t / ms
|hc (φ, t)| Continuous measurements (145–167 ms) 166
−60
164
−65
162
−70
160
−75
158
−80
156
−85
154
−90
152
−95
150
−100
148
−105
146
−110 −150 −100 −50
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0 50 φ / deg
Continuous IR Measurement
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| Measurements
150
dB
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t / ms
|hs (φ, t)| Sequential measurements (145–167 ms) 166
−60
164
−65
162
−70
160
−75
158
−80
156
−85
154
−90
152
−95
150
−100
148
−105
146
−110 −150 −100 −50
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0 50 φ / deg
Continuous IR Measurement
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| Measurements
150
dB
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Comparison root mean square (RMS) difference
s
N−1 P
ε(φ) =
2 hs (φ, k) − hc (φ, k) k=0 s 2 N−1 P hs (φ, k) k=0
cross-correlation (after upsampling by a factor of 10) N−1 P
hs (φ, k)hc (φ, k + κ) s ρsc (φ, κ) = s 2 N−1 2 N−1 P P hs (φ, k) hc (φ, k) k=−N+1
k=0
maxκ {ρsc (φ, κ)} arg maxκ {ρsc (φ, κ)} N.Hahn and S.Spors
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k=0
similarity of the waveforms time misalignment
Continuous IR Measurement
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RMS Difference
ε / dB
0
−5
−10
−15
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−150
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−100
|
−50
Continuous IR Measurement
0 φ/◦
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| Measurements
100
150
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Impulse Responses
φ = 20◦ , ε = −14.2 dB
Amplitude / dB
−20
seq. cont.
−40 −60 −80 3.5
4
4.5
5
5.5
6
6.5
t / ms N.Hahn and S.Spors
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Frequency Responses φ = 20◦ , ε = −14.2 dB seq. cont. diff.
Magnitude / dB
0 −10 −20 −30 102
103
104
f / Hz
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Impulse Responses
φ = −50◦ , ε = −2.1 dB
Amplitude / dB
−20
seq. cont.
−40 −60 −80 4
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|
5.5 t / ms
Continuous IR Measurement
| Measurements
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6.5
7
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Frequency Responses φ = −50◦ , ε = −2.1 dB seq. cont. diff.
Magnitude / dB
0 −10 −20 −30 102
103
104
f / Hz
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Maximum Cross-correlation ρsc ρ¯ss
ρmax cs
1
0.98
0.96
−150
−100
−50
0 φ/◦
50
100
150
ρ¯ss = 0.984 mean value of ρss (φ, κ) computed from 720 measurements of hs (φ, k) N.Hahn and S.Spors
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Time Lag τsc ±σsc
τ / µs
10
0 −10 −150
−100
−50
0 φ/◦
50
100
150
σsc = 0.528 µs standard deviation of τss (φ, κ) computed from 720 measurements of hs (φ, k) N.Hahn and S.Spors
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RMS Difference Reduction after Time Alignment
ε / dB
0
original aligned
−5
−10
−15
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−150
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−100
|
−50
Continuous IR Measurement
0 φ/◦
50
| Measurements
100
150
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Discussion
Informal listening speech, castanets, and pink noise signals filtered with individual impulse responses (hs (φ, k) and hs (φ, k)) indistinguishable Practical Issues vibrational and frictional noise accurate control of Ωφ synchronization of motor control and audio capturing
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| Measurements
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Conclusion
Summary continuous measurement technique for room impulse responses maximum allowable Ωφ derived based on the spatial bandwidth of h(φ, t) comparison of continuous and sequential measurements on a circle Future work perceptual evaluation in the context of data-based sound reproduction extension to three-dimensional case, e.g. continuous measurement on a sphere
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Continuous IR Measurement
| Conclusion
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