frequency domain coding of speech
DESCRIPTION
Frequency Domain Coding of Speech. 主講人:虞台文. Content. Introduction The Short-Time Fourier Transform The Short-Time Discrete Fourier Transform Wide-Band Analysis/Synthesis Sub-Band Coding. Frequency Domain Coding of Speech. Introduction. Speech Coders. Waveform Coders - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/1.jpg)
Frequency Domain Coding of Speech
主講人:虞台文
![Page 2: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/2.jpg)
Content Introduction The Short-Time Fourier Transform The Short-Time Discrete Fourier Transform Wide-Band Analysis/Synthesis Sub-Band Coding
![Page 3: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/3.jpg)
Frequency Domain Coding of Speech
Introduction
![Page 4: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/4.jpg)
Speech Coders Waveform Coders
– Attempt to reproducing the original waveform according to some fidelity criteria
– Performance: successful at producing good quality, robust speech.
Vocoders– Correlated with speech production model.– Performance: more fragile and more model depend
ent.– Lower bit rate
![Page 5: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/5.jpg)
Frequency-Domain Coders
Sub-band coder (SCB). Adaptive Transform Coding (ATC). Multi-band Excited Vocoder (MBEV). Noise Shaping in Speech Coders.
![Page 6: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/6.jpg)
Classification of Speech Coders
![Page 7: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/7.jpg)
Frequency Domain Coding of Speech
The Short-Time Fourier Transform
![Page 8: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/8.jpg)
Definition of STFT
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
Interpretations:Filter Bank InterpretationBlock Transform Interpretation
![Page 9: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/9.jpg)
Filter Bank Interpretation
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
is fixed at 0.
])([*)()( 00 njjn enxnheX
f (m)AnalysisFilter
![Page 10: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/10.jpg)
Filter Bank Interpretation
...
nje 1
nje 2
nj Me 1
nj Me
)( 1jn eX
)( 2jn eX
)( 3jn eX
)( 4jn eX
h(n)h(n)
h(n)h(n)
h(n)h(n)
h(n)h(n)
x(n)
])([*)()( 00 njjn enxnheX
![Page 11: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/11.jpg)
Filter Bank Interpretation
])([*)()( 00 njjn enxnheX
Modulation
)( 00)( jFTnj eXenx )( 00)( jFTnj eXenx
)( jeX )(nx
nje 0
)(nx
)( 0)( jj eXeX
0
])([*)()( 00 njjn enxnheX
![Page 12: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/12.jpg)
Filter Bank Interpretation
)( jeX )(nx
nje 0
)(nx
)( 0)( jj eXeX
0
LowpassFilter
])([*)()( 00 njjn enxnheX
Modulation
![Page 13: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/13.jpg)
Filter Bank Interpretation
])([*)()( 00 njjn enxnheX
...
nje 1
nje 2
nj Me 1
nj Me
)( 1jn eX
)( 2jn eX
)( 3jn eX
)( 4jn eX
h(n)h(n)
h(n)h(n)
h(n)h(n)
h(n)h(n)
x(n) Modulated Subband signals
![Page 14: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/14.jpg)
Block Transform Interpretation
m
mjjn emxmnheX )()()( 00
m
mjjn emxmnheX )()()( 00
n is fixed at n0.
Windowed Data
AnalysisWindow
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
FT of Windowed Data
)]()([)( 00nxnnhFTeX j
n )]()([)( 00nxnnhFTeX j
n
![Page 15: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/15.jpg)
Block Transform Interpretation
n is fixed at n0. )]()([)( 00nxnnhFTeX j
n )]()([)( 00nxnnhFTeX j
n
n1
n2
n3...nr
)(1
jn eX )(1
jn eX
)(2
jn eX )(2
jn eX
)(3
jn eX )(3
jn eX
)( jn eX
r
)( jn eX
r
![Page 16: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/16.jpg)
Analysis/Synthesis Equations
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(Analysis
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆSynthesis
In what condition we will have ?)(ˆ)( nxnx
![Page 17: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/17.jpg)
Analysis/Synthesis Equations
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(Analysis
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆSynthesis
deeXrnfnx njjr
r
)(2
1)()(ˆ )()()( nxnrhrnf
r
)()()( nrhrnfnxr
Replace r with n+r
)()()( rhrfnxr
![Page 18: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/18.jpg)
Analysis/Synthesis Equations
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(Analysis
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆSynthesis
deeXrnfnx njjr
r
)(2
1)()(ˆ )()()( nxnrhrnf
r
)()()( nrhrnfnxr
Therefore, )(ˆ)( nxnx if 1)()(
nhnfn
1)()(
nhnfn
)()()( rhrfnxr
![Page 19: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/19.jpg)
Analysis/Synthesis Equations
More general, 1)()(2
1)()(
deHeFnhnf jj
n
1)()(2
1)()(
deHeFnhnf jj
n
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(Analysis
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆSynthesis
Therefore, )(ˆ)( nxnx if 1)()(
nhnfn
1)()(
nhnfn
![Page 20: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/20.jpg)
Examples1)()(
2
1)()(
deHeFnhnf jj
n
1)()(2
1)()(
deHeFnhnf jj
n
0)0( ,)0(
)()(
h
h
nnf 1)()(
nhnfn
neH
nfj
allfor ,)(
1)(
0
)(
)()(
0jj
eHeF
1( ) ( ) 1
2j jF e H e d
![Page 21: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/21.jpg)
Examples
0)0( ,)0(
)()(
h
h
nnf
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆ
deeXh
nx njjn )(
2
1
)0(
1)(ˆ
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
h(0)x(n)
)(nx
![Page 22: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/22.jpg)
Examples
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjrj
deeXeH
nx )(2
1
)(
1)(ˆ
0
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
neH
nfj
allfor ,)(
1)(
0
j
n
j enheH )()(
n
j nheH )()( 0
r
jr
r
eXFTrh
)]([)(
1 1
r
r
nxnrhrh
)()()(
1
r
r
nxrhrh
)()()(
1)(nx
![Page 23: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/23.jpg)
Frequency Domain Coding of Speech
The Short-Time Discrete Fourier Transform
![Page 24: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/24.jpg)
Definition of STDFT
m
kmM
Mkmjnn WmxmnheXkX )()(][)( )/2(
m
kmM
Mkmjnn WmxmnheXkX )()(][)( )/2(
Analysis:
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
Synthesis: In what condition we will have?)(ˆ)( nxnx
r
njjr deeXrnfnx )()(
2
1)(ˆ
r
njjr deeXrnfnx )()(
2
1)(ˆ
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
)/2( MjM eW
)/2( MjM eW
![Page 25: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/25.jpg)
Synthesis
1
0
)(1
)()(ˆM
k
knMr
r
WkXM
rnfnx
m
kmMn WmxmnhkX )()()(
m
kmMn WmxmnhkX )()()(
)()()()(ˆ nxnrhrnfnxr
)()()( nrhrnfnxr
1)(nx
1)()(
nrhrnfr
1)()(
nrhrnfr
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
![Page 26: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/26.jpg)
Synthesis
1
0
)(1
)()(ˆM
k
knMr
r
WkXM
rnfnx
)()()()(ˆ nxnrhrnfnxr
)()()( nrhrnfnxr
)(nx
1)()(
nrhrnfr
1)()(
nrhrnfr
periodic. are )()(ˆBoth nxnx periodic. are )()(ˆBoth nxnx
)()(
)(ˆ)(ˆ
Mnxnx
Mnxnx
)()(
)(ˆ)(ˆ
Mnxnx
Mnxnx
We need only one period.
Therefore, the condition is respecified as:
)()]([)( ppMnrhrnfr
)()]([)( ppMnrhrnfr
![Page 27: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/27.jpg)
Implementation Consideration
n
Fre
quen
cyk
0Spectrogram
![Page 28: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/28.jpg)
Sampling
n
Fre
quen
cyk
0Spectrogram
R 2R 3R 4R
)(0 kX R)(0 kX R )(kX R
)(kX R )(2 kX R)(2 kX R )(3 kX R
)(3 kX R )(4 kX R)(4 kX R
![Page 29: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/29.jpg)
Sampled STDFT
m
kmMn WmxmnhkX )()()(
m
kmMn WmxmnhkX )()()(
Analysis:
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
Synthesis: In what condition we will have?)(ˆ)( nxnx
m
kmMsR WmxmsRhkX )()()(
m
kmMsR WmxmsRhkX )()()(
1
0
)()(1
)(ˆM
k s
knMsR WkXsRnf
Mnx
1
0
)()(1
)(ˆM
k s
knMsR WkXsRnf
Mnx
![Page 30: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/30.jpg)
Sampled STDFT
m
kmMn WmxmnhkX )()()(
m
kmMn WmxmnhkX )()()(
Analysis:
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
1
0
)()(1
)(ˆM
k r
knMr WkXrnf
Mnx
Synthesis: In what condition we will have?)(ˆ)( nxnx
m
kmMsR WmxmsRhkX )()()(
m
kmMsR WmxmsRhkX )()()(
1
0
)()(1
)(ˆM
k s
knMsR WkXsRnf
Mnx
1
0
)()(1
)(ˆM
k s
knMsR WkXsRnf
Mnx
)()]([)( ppMnrhrnfr
)()]([)( ppMnrhrnfr
)()]([)( ppMnsRhsRnfs
)()]([)( ppMnsRhsRnfs
![Page 31: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/31.jpg)
Frequency Domain Coding of Speech
Wide-Band
Analysis/Synthesis
![Page 32: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/32.jpg)
Short-Time Synthesis --- Filter Bank Summation
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
m
mjjn
kk emxmnheX )()()(
STFT
h(n)h(n)x(n)
nj ke
)( kjn eX
nj kenxnh )(*)(
LowpassFilter
![Page 33: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/33.jpg)
Short-Time Synthesis --- Filter Bank Summation
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
m
nmjjn
kk emhmnxeX )()()()(
STFT
m
mjnj kk emhmnxe )()(
m
knjj
n mhmnxeeX kk )()()(
m
knjj
n mhmnxeeX kk )()()(nj
kkenhnh )()(nj
kkenhnh )()(
![Page 34: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/34.jpg)
Short-Time Synthesis --- Filter Bank Summation
|H(ej)|
|Hk(ej)|
k
Lowpass filter Bandpass filter
( )( ) kjjkH e H e ( )( ) kjj
kH e H e
m
knjj
n mhmnxeeX kk )()()(
m
knjj
n mhmnxeeX kk )()()(nj
kkenhnh )()(nj
kkenhnh )()(
![Page 35: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/35.jpg)
Short-Time Synthesis --- Filter Bank Summation
hk(n)hk(n)x(n))( kj
n eX
BandpassFilter nj ke
m
mjjn emxmnheX )()()(
m
mjjn emxmnheX )()()(
h(n)h(n)x(n)
nj ke
)( kjn eX
LowpassFilter
Lowpass representation of for the signal in a band centered at k.
m
knjj
n mhmnxeeX kk )()()(
m
knjj
n mhmnxeeX kk )()()(nj
kkenhnh )()(nj
kkenhnh )()(
![Page 36: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/36.jpg)
Short-Time Synthesis --- Filter Bank Summation
hk(n)hk(n)x(n))( kj
n eX
BandpassFilter nj ke
h(n)h(n)x(n)
nj ke
)( kjn eX
LowpassFilter
nj ke
)(nyk
nj ke
)(nyk
Encoding one band Decoding one band
)(*)()()( nhnxeeXny knjj
nkkk )(*)()()( nhnxeeXny knjj
nkkk
![Page 37: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/37.jpg)
Short-Time Synthesis --- Filter Bank Summation
)(*)()()( nhnxeeXny knjj
nkkk )(*)()()( nhnxeeXny knjj
nkkk
h1(n)h1(n))( 1j
n eX
)(1 ny
nje 1 nje 1x(n)
nje 0
h0(n)h0(n))( 0j
n eX )(0 nynje 0
hN1(n)hN1(n))( 1Nj
n eX
)(1 nyN
nj Ne 1 nj Ne 1
.
.
.
)(ny
AnalysisAnalysis SynthesisSynthesis
![Page 38: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/38.jpg)
Short-Time Synthesis --- Filter Bank Summation
h1(n)h1(n))( 1j
n eX
)(1 ny
nje 1 nje 1x(n)
nje 0
h0(n)h0(n))( 0j
n eX )(0 nynje 0
hN1(n)hN1(n))( 1Nj
n eX
)(1 nyN
nj Ne 1 nj Ne 1
.
.
.
)(ny
AnalysisAnalysis SynthesisSynthesis
![Page 39: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/39.jpg)
Short-Time Synthesis --- Filter Bank Summation
h1(n)h1(n))( 1j
n eXnje 1 nje 1
x(n)
nje 0
h0(n)h0(n))( 0j
n eX )(0 nynje 0
hN1(n)hN1(n))( 1Nj
n eX
)(1 nyN
nj Ne 1 nj Ne 1
.
.
.
)(ny
AnalysisAnalysis SynthesisSynthesis
)(1 ny
)()( )( kjjk eHeH )()( )( kjj
k eHeH
![Page 40: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/40.jpg)
Equal Spaced Ideal Filters
N2
N2
N2
N2
N2
N2
N2
1 2 3 4 5 21 0
N = 6
)()( )( kjjk eHeH )()( )( kjj
k eHeH N
kk
2N
kk
2
![Page 41: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/41.jpg)
Equal Spaced Ideal Filters
)(0 ny
)(1 nyN
)(ny)(1 nyh1(n)
x(n)
h0(n)
hN1(n)
.
.
.
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
What condition should be satisfied so that y(n)=x(n)?
)()( )( kjjk eHeH )()( )( kjj
k eHeH N
kk
2N
kk
2
![Page 42: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/42.jpg)
Equal Spaced Ideal Filters
)()( )( kjjk eHeH )()( )( kjj
k eHeH N
kk
2N
kk
2
1
0
)(1 N
k
njj kk eeHN
r
rNnh )(
Equal spaced sampling of
H(ej )
Inverse discrete FT of H(ej )
Time-Aliasedversion of h(n)
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
![Page 43: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/43.jpg)
Equal Spaced Ideal Filters
)()( )( kjjk eHeH )()( )( kjj
k eHeH N
kk
2N
kk
2
1
0
)(1 N
k
njj kk eeHN
r
rNnh )(
Consider FIR, i.e., h(n) is of duration of L samples.
0 L1 n
h(n)
In case that N L,
1
0
)0()(1 N
k
j heHN
k
1
0
)0()(1 N
k
j heHN
k
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
![Page 44: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/44.jpg)
Equal Spaced Ideal Filters
)()( )( kjjk eHeH )()( )( kjj
k eHeH N
kk
2N
kk
2
1( )
0
( ) ( )k
Njj
k
H e H e
1
0
( )k
Nj
k
H e
)0(Nh
1
0
)0()(1 N
k
j heHN
k
1
0
)0()(1 N
k
j heHN
k
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
![Page 45: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/45.jpg)
Equal Spaced Ideal Filters
)0()(~
NheH j )0()(~
NheH j
)(0 ny
)(1 nyN
)(ny)(1 nyh1(n)h1(n)
x(n)
h0(n)h0(n)
hN1(n)hN1(n)
.
.
.
)()0()( nxNhny )()0()( nxNhny
0 L1 n
h(n)
x(n) can always beReconstructed if N L,
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
![Page 46: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/46.jpg)
Equal Spaced Ideal Filters
)0()(~
NheH j )0()(~
NheH j
)(0 ny
)(1 nyN
)(ny)(1 nyh1(n)h1(n)
x(n)
h0(n)h0(n)
hN1(n)hN1(n)
.
.
.
0 L1 n
h(n)
x(n) can always beReconstructed if N L,
Does x(n) can still be reconstructed if N<L?Does x(n) can still be reconstructed if N<L?
If affirmative, what condition should be satisfied?If affirmative, what condition should be satisfied?
)()0()( nxNhny )()0()( nxNhny
1
0
)()(~ N
k
jk
j eHeH
1
0
)()(~ N
k
jk
j eHeH
![Page 47: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/47.jpg)
Equal Spaced Ideal Filters
)(0 ny
)(1 nyN
)(ny)(1 nyh1(n)h1(n)
x(n)
h0(n)h0(n)
hN1(n)hN1(n)
.
.
.
njk
kenhnh )()(nj
kkenhnh )()(
njN
k
kenhnh
1
0
)()(~
N
kk
2N
kk
2
1
0
)(N
k
nj kenh
p(n)
r
rNnNnp )()(
r
rNnNnp )()(
![Page 48: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/48.jpg)
Equal Spaced Ideal Filters
njN
k
kenhnh
1
0
)()(~
1
0
)(N
k
nj kenh
p(n)
r
rNnNnp )()(
r
rNnNnp )()(
)()()(~
npnhnh
r
rNnrNhN )()(
Signal can be reconstructedIf it equals to (n m).
)()()(~
npnhnh )()()(~
npnhnh
r
rNnnNh )()(
![Page 49: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/49.jpg)
Typical Sequences of h(n)
)()()(~
npnhnh )()()(~
npnhnh
Ideal lowpass filter with cutoff at /N.
n
nnh N
sin)(
n
nnh N
sin)(
0N2N N 2N 3N 4N
p(n)N
)()(~
nnh )()(~
nnh
0N2N N 2N 3N 4N
h(n)
1/N
![Page 50: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/50.jpg)
Typical Sequences of h(n)
)()()(~
npnhnh )()()(~
npnhnh
0N2N N 2N 3N 4N
p(n)N
0N2N N 2N 3N 4N
h(n)
h(0)
)()0()(~
nNhnh )()0()(~
nNhnh
L2L L 2L 3L 4L
N L
![Page 51: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/51.jpg)
Typical Sequences of h(n)
)()()(~
npnhnh )()()(~
npnhnh
0N2N N 2N 3N 4N
p(n)N
)2()(~
Nnnh )2()(~
Nnnh
0N2N N 2N 3N 4N
h(n)
h(0)
1/N A causalFIR lowpass filter
![Page 52: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/52.jpg)
Typical Sequences of h(n)
)()()(~
npnhnh )()()(~
npnhnh
0N2N N 2N 3N 4N
p(n)N
)()(~
Nnnh )()(~
Nnnh
0N2N N 2N 3N 4N
h(n)
h(0)
1/N A causalIIR lowpass filter
![Page 53: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/53.jpg)
Filter Back Implementation for a Single Channel
hk(n)x(n))( kj
n eX
nj ke nj ke
)(nyk
h(n)x(n)
nj ke
)( kjn eX
nj ke
)(nyk
AnalysisAnalysis SynthesisSynthesis
![Page 54: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/54.jpg)
hk(n)x(n))( kj
n eX
nj ke nj ke
)(nyk
h(n)x(n)
nj ke
)( kjn eX
nj ke
)(nyk
Filter Back Implementation for a Single Channel
R:1
R:1
1:R
1:R)( kj
n eX
)( kjn eX
AnalysisAnalysis SynthesisSynthesis
DecimatorDecimator InterpolatorInterpolator
![Page 55: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/55.jpg)
hk(n)x(n))( kj
n eX
nj ke nj ke
)(nyk
h(n)x(n)
nj ke
)( kjn eX
nj ke
)(nyk
Filter Back Implementation for a Single Channel
R:1
R:1
1:R
1:R)( kj
n eX
)( kjn eX
AnalysisAnalysis SynthesisSynthesis
DecimatorDecimator InterpolatorInterpolator
Depends on the bandwidth of h(n).Depends on the bandwidth of h(n).
R=?R=?
![Page 56: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/56.jpg)
Frequency Domain Coding of Speech
Sub-Band Coding
![Page 57: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/57.jpg)
AnalysisAnalysis SynthesisSynthesis
Filter Bank Implementation(Direct Implementation)
...
0NW
h(n)h(n)
h(n)h(n)
h(n)h(n)
h(n)h(n)
x(n)n
NW
knNW
nNNW )1(
...
)0(sRXR:1R:1
R:1R:1
R:1R:1
R:1R:1
)1(sRX
)(kX sR
)1( NX sR
1:R1:R
1:R1:R
1:R1:R
1:R1:R
...
...
f(n)f(n)
f(n)f(n)
f(n)f(n)
f(n)f(n)
0NW
nNW
knNW
nNNW )1(
x(n)
Complex ChannelsComplex Channels R=2BR=2B
Bandwidth B/2
![Page 58: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/58.jpg)
Filter Bank Implementation(Practical Implementation)
0
B
k0
B
k
0 B/2B/2 0 B/2B/2
0B 0 B
0B B
knNW kn
NW knNWkn
NW
2/jBne2/jBne 2/jBne
2/jBne
![Page 59: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/59.jpg)
Filter Bank Implementation(Practical Implementation)
)()()( njbnaeX kkj
nk
)()()( njbnaeX kkj
nk
...
...
h(n)h(n)
h(n)h(n)
x(n)
knNW
knNW
...2/jBne
2/jBne
)(nyk
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
![Page 60: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/60.jpg)
Filter Bank Implementation(Practical Implementation)
)2/cos(Bn
)(21 nyk
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
)2/sin(Bn
)(nak
)(nbk
nkcos
nksin
...
h(n)h(n)
x(n)
...
h(n)h(n)
![Page 61: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/61.jpg)
)(21 sDyk
)2/cos(BsD
)2/sin(BsD
)(nak
)(nbk
nkcos
nksin
...
h(n)h(n)
x(n)
...
h(n)h(n)
Filter Bank Implementation(Practical Implementation)
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
D:1D:1
D:1D:1
BD / BD /
Why?
)(sDak
)(sDbk
![Page 62: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/62.jpg)
Filter Bank Implementation(Practical Implementation)
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
)(21 sDyk
)2/cos(BsD
)2/sin(BsD
)(nak
)(nbk
nkcos
nksin
...
h(n)h(n)
x(n)
...
h(n)h(n)
D:1D:1
D:1D:1
BD / BD /)(sDak
)(sDbk
)2/cos( s )2/cos( s
)2/sin( s )2/sin( s
![Page 63: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/63.jpg)
)(21 sDyk
)(sDak
)(sDbk
)(nak
)(nbk
nkcos
nksin
...
h(n)h(n)
x(n)
...
h(n)h(n)
)2/cos( s )2/cos( s
)2/sin( s )2/sin( s
D:1D:1
D:1D:1
Filter Bank Implementation(Practical Implementation)
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
,0,1,0,1,0,1 ,0,1,0,1,0,1
,1,0,1,0,1,0 ,1,0,1,0,1,0
![Page 64: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/64.jpg)
s)1(
Filter Bank Implementation(Practical Implementation)
)2/sin()(2)2/cos()(2)( BnnbBnnany kkk )2/sin()(2)2/cos()(2)( BnnbBnnany kkk
s)1(
)2( Dsak
)2( Dsbk
x(n)
)(nak
)(nbk
nkcos
nksin
...
h(n)h(n)
...
h(n)h(n)
)(21 sDyk
D:1D:1
D:1D:1
2D:12D:1
2D:12D:1
![Page 65: Frequency Domain Coding of Speech](https://reader035.vdocuments.site/reader035/viewer/2022062408/568134d1550346895d9bf84d/html5/thumbnails/65.jpg)
Filter Bank Implementation(Practical Implementation)
ADPCMCODEC
s)1(
s)1(
)2( Dsak
)2( Dsbk
nkcos
nksin...
h(n)h(n)
...
h(n)h(n)
2D:12D:1
2D:12D:1
)(nx
f(n)f(n)
...
f(n)f(n)
2D:12D:1
2D:12D:1
s)1(
s)1(
nkcos
nksin...
)2(ˆ Dsak
)2(ˆ Dsbk
)(ˆ nxk
Filter BankAnalysis
Filter BankAnalysis Sub-Band Coder
ModificationSub-Band Coder
Modification Filter BankSynthesis
Filter BankSynthesis