We should use causal reconstruction. ) (sinc)(. )( ~ nht nhf tf n. â. = â. â. ââ. = ). (][. )( ~. 0 nhtng tf n. â. = â. â. = Ï support compact aith function w hold: filter.
YY filter A Paradigm of Digital Signal Processing Masaaki Nagahara Kyoto University
Table of contents Shannon
sampling theorem and its problems YY filter (a paradigm of signal reconstruction) Application Digital
synthesis of musical instruments
Examples Conclusion
Mathematics for Systems and Control, Yutaka Yamamoto, 1998. The
first book by which I learned a strict mathematics.
Mathematics for Systems and Control, Yutaka Yamamoto, 1998. The
first book by which I learned a strict mathematics. And also the first book by which I learned Shannon sampling theorem with a mathematical proof.
Shannon Sampling Theorem
Assume that f (t ) is a continuous function in S ′ (the set of tempered distributions), and the Fourier transform fˆ = Ff is fully band-limited by supp fˆ ⊂ (−π / h, π / h).
Then we have f (t ) =
∞
∑
f (nh)
n = −∞
=
∞
∑ f (nh)sinc(t − nh)
n = −∞
sinc(t ) =
sin π (t / h − n) π (t / h − n)
sin π (t / h) π (t / h)
Shannon Sampling Theorem For
the band-limited signals (BL signals), the sampled signals can be exactly reconstructed by the inversion formula (Fourier series expansion in terms of sinc functions).
{
}
B = f ∈ S ′ : supp fˆ ∈ (−π / h, π / h) ∞
f ∈B
S
f (nh)
* S
∑ f (nh)sinc(t − nh)
n = −∞
Problems in Shannon Reconstruction
Band limiting assumption:
{
}
f ∈ B = f ∈ S ′ : supp fˆ ∈ (−π / h, π / h)
Real signals such as audio signals are never band-limited.
fˆ ( jω )
ω −π / h
π /h
Problems in Shannon Reconstruction
Band limiting assumption:
{
}
f ∈ B = f ∈ S ′ : supp fˆ ∈ (−π / h, π / h)
Real signals such as audio signals are never band-limited.
More reasonable space is the set of filtered L2 signals by a stable and strictly causal LTI filter F :
{
f ∈ FL2 = f ∈ L2 : f = Fw, w ∈ L2
}
Problems in Shannon Reconstruction
Causality: ~ f (t ) =
∞
∑ f (nh)sinc(t − nh)
n = −∞
To reconstruct f at time t, one needs whole data of {f (nh): n=...,-2,-1,0,1,2,...}.
We should use causal reconstruction. ∞ ~ f (t ) = ∑ g[n]φ (t − nh) n =0
g : filtered { f (nh)} by a causal filter K
φ : hold function with a compact support
YY Filter the filter K(z) which minimizes the H∞ norm (L2-induced norm) of the error system:
Find
f ∈ FL2
f (nh), n = 0,1,2,...
fˆ
w ∈ L2 [0, ∞) F(s) Causal Stable LTI (LPF)
S
K(z)
ZOH
e = f (• − L) − fˆ
－
e－Ls
This Thiscan canbe besolved solvedby bysampled-data sampled-datacontrol controltheory theory!!
Applications of YY filter Multirate
signal processing
Oversampling
AD/DA converters Sampling rate converters Audio compression Image zooming Probabilistic Estimating
Digital
density estimation
density from coarse histogram
repetitive control Digital sound synthesis
Applications of YY filter Multirate
signal processing
Oversampling
AD/DA converters Sampling rate converters Audio compression Image zooming Probabilistic Estimating
Digital
density estimation
density from coarse histogram
repetitive control Digital sound synthesis
Digital sound synthesis Playing
musical instruments in a computer (Computer Music) This requires musical instruments installed in computer digitally (Digital sound synthesis)
Existing Methods FM
(Frequency Modulation) synthesis
Sounds
are synthesized by linear and nonlinear combination of simple waveforms (sin, triangle, rectangle, etc…) with frequency modulation. Requires small memory, but the design is hard. PCM
(Pulse Code Modulation) synthesis
Sounds
are synthesized by just recording. The design is easy, but requires large memory.
New Method Digitally
record a single tone (e.g., A or la, 440Hz) of a musical instrument. Shift the pitch to create another tone.
1/440 Pitch shifting
1/440×22/12
Pitch Shifting in Digital The
original tone is recorded digitally.
Pitch Shifting in Digital The
original tone is recorded digitally.
Pitch
shifted tone with equal sampling frequency should be
Pitch Shifting in Digital The
original tone is recorded digitally.
Pitch
shifted tone with equal sampling frequency should be
Pitch Shifting in Digital The
original tone is recorded digitally.
period shifted tone withSampling equal sampling We have to estimate this value from frequency should be the original sampled data.
Pitch
Pitch Shifting in Digital The
value to be estimated is obtained by (virtually) shifting the original curve and sampling the curve.
Sampling period
Pitch Shifting in Digital The
value to be estimated is obtained by (virtually) shifting the original curve and sampling the curve.
Sampling period
Pitch Shifting in Digital The
value to be estimated is obtained by (virtually) shifting the original curve and sampling the curve.
Sampling period
Pitch Shifting in Digital The
value to be estimated is obtained by (virtually) shifting the original curve and sampling the curve. This must be done in digital. Fractional delay filters can do this. Sampling period
Fractional Delay Filters Find
the optimal filter K(z) which minimizes the H∞ norm of the error system: Virtual delay
Signal to be estimated
w ∈ L2 [0, ∞) F(s)
F(s) is chosen according to the characteristic of musical instrument
e－Ls
S
S
K(z)
e －
Estimated signal Recorded signal
The Optimal Fractional Delay Filter We
assume that F(s) is a 1st order LPF: ωc F (s) = s + ωc
Sampling
period: h Delay: L = mh + d (m: integer, d < h) Then the optimal filter is given by
(
K ( z ) = z − m a0 + a1 z −1
) (e
sinh ωc (h − d ) , a1 = e −ωc h a0 = ωc sinh ωc h
ωc d
− a0
)
Example Recording
an A (440Hz) tone by playing the
guitar.
Shift
the pitch to produce other tones by the optimal fractional delay filter. C#
E
B
Example Music
produced by one tone.
Conclusion YY
filter is the optimal digital filter which minimizes the H∞ norm of the sampled-data error system taking account of analog characteristic. YY filter has many application in digital signal processing. Digital
And…
sound synthesis
Happy birthday Yamamoto Sensei !
