1 Digital Speech Processing Digital Speech Processing— Lecture 2 Lecture 2 1 Review of DSP Review of DSP Fundamentals Fundamentals What is DSP? What is DSP? Analog-to- Digital Conversion Computer Input Signal Output Signal Digital-to- Analog Conversion Digital • Method to represent a quantity, a phenomenon or an event 2 • Why digital? Signal • What is a signal? – something (e.g., a sound, gesture, or object) that carries information – a detectable physical quantity (e.g., a voltage, current, or magnetic field strength) by which messages or information can be transmitted • What are we interested in, particularly when the signal is speech? Processing • What kind of processing do we need and encounter almost everyday? • Special effects? Common Forms of Computing Common Forms of Computing • Text processing – handling of text, tables, basic arithmetic and logic operations (i.e., calculator functions) – Word processing – Language processing – Spreadsheet processing – Presentation processing 3 Presentation processing • Signal Processing – a more general form of information processing, including handling of speech, audio, image, video, etc. – Filtering/spectral analysis – Analysis, recognition, synthesis and coding of real world signals – Detection and estimation of signals in the presence of noise or interference Advantages of Digital Representations Advantages of Digital Representations • Permanence and robustness of signal representations; zero- Signal Processor Input Signal Output Signal A-to-D Converter D-to-A Converter 4 distortion reproduction may be achievable • Advanced IC technology works well for digital systems • Virtually infinite flexibility with digital systems – Multi-functionality – Multi-input/multi-output • Indispensable in telecommunications which is virtually all digital at the present time Digital Processing of Analog Signals Digital Processing of Analog Signals • A-to-D conversion: bandwidth control, sampling and quantization • Computational processing: implemented on computers or x c (t) x[n] y[n] y c (t) A-to-D Computer D-to-A 5 Computational processing: implemented on computers or ASICs with finite-precision arithmetic – basic numerical processing: add, subtract, multiply (scaling, amplification, attenuation), mute, … – algorithmic numerical processing: convolution or linear filtering, non-linear filtering (e.g., median filtering), difference equations, DFT, inverse filtering, MAX/MIN, … • D-to-A conversion: re-quantification* and filtering (or interpolation) for reconstruction Discrete Discrete-Time Signals Time Signals { [ ]}, ( ), , [] ( ) A sequence of numbers Mathematical representation: Sampled from an analog signal, at time a x xn n x t t nT xn x nT n = −∞< <∞ = = − ∞< <∞ i i i 6 [] ( ), is called the and its recipr a xn x nT n T = ∞< <∞ sampling period, i 1/ , 8000 1 / 8000 125 sec 10000 1/10000 100 sec 16000 1/16000 62.5 sec 40000 1 / 40000 25 sec ocal, is called the Hz Hz Hz Hz S S S S S F T F T F T F T F T μ μ μ μ = = ↔ = = = ↔ = = = ↔ = = = ↔ = = sampling frequency
Transcript
1
Digital Speech ProcessingDigital Speech Processing——Lecture 2Lecture 2
1
Review of DSP Review of DSP FundamentalsFundamentals
What is DSP?What is DSP?
Analog-to-Digital
ConversionComputer
InputSignal
OutputSignalDigital-to-
Analog Conversion
Digital• Method to represent a quantity, a phenomenon or an event
Wh di it l?
2
• Why digital?Signal
• What is a signal?– something (e.g., a sound, gesture, or object) that carries information– a detectable physical quantity (e.g., a voltage, current, or magnetic field strength) by
which messages or information can be transmitted • What are we interested in, particularly when the signal is speech?
Processing• What kind of processing do we need and encounter almost
everyday?• Special effects?
Common Forms of ComputingCommon Forms of Computing• Text processing – handling of text, tables, basic
arithmetic and logic operations (i.e., calculator functions)– Word processing– Language processing– Spreadsheet processing– Presentation processing
3
Presentation processing
• Signal Processing – a more general form of information processing, including handling of speech, audio, image, video, etc.– Filtering/spectral analysis– Analysis, recognition, synthesis and coding of real world signals– Detection and estimation of signals in the presence of noise or
interference
Advantages of Digital RepresentationsAdvantages of Digital Representations
• Permanence and robustness of signal representations; zero-
Signal Processor
InputSignal
OutputSignalA-to-D
ConverterD-to-A
Converter
4
distortion reproduction may be achievable• Advanced IC technology works well for digital systems• Virtually infinite flexibility with digital systems
– Multi-functionality– Multi-input/multi-output
• Indispensable in telecommunications which is virtually all digital at the present time
Digital Processing of Analog SignalsDigital Processing of Analog Signals
• A-to-D conversion: bandwidth control, sampling and quantization
• Computational processing: implemented on computers or
xc(t) x[n] y[n] yc(t)A-to-D Computer D-to-A
5
Computational processing: implemented on computers or ASICs with finite-precision arithmetic– basic numerical processing: add, subtract, multiply
(scaling, amplification, attenuation), mute, …– algorithmic numerical processing: convolution or linear
[ ]x ni can be quantized to one of a finite set of values which isthen represented digitally in bits, hence a truly digital signal; thecourse material mostly deals with discrete-time signals (discrete-value only when noted).
in0.3 0.9 1.5 2.1
1.8
1.2
0.6
6
5
4
3
2
1
0
A 3-bit uniform quantizer
valued input into a representation that assumes one out of a finite set of values
• The finite set of output values is indexed; e.g., the value 1.8 has an index of 6, or (110)2 in binary representation
• Storage or transmission uses binary representation; a quantization table is needed
2
4
6
-6
-4
-2
0
2
4
6
y
2
4
6
Discrete SignalsDiscrete Signals
Analog sinusoid, 5sin(2πx)
Sampled Sinusoid 5sin(2πnT)
sample quantize
Discrete sinusoid
round[5sin(2πnT)]
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
x
y
0 5 10 15 20 25 30 35 406
n
0 5 10 15 20 25 30 35 40-6
-4
-2
0
n
y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
x
y
( )
Quantized sinusoid
round[5sin(2πx)]
samplequantize
SinewaveSinewave SpectrumSpectrum
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SNR is a function of B, the number of bits in the quantizer
3
Issues with Discrete SignalsIssues with Discrete Signals• what sampling rate is appropriate
• A bandlimited signal can be reconstructed exactly from samples taken with sampling frequency
0 0.2 0.4 0.6 0.8 1 1.2
-1
time in ms
max max1 22 2 or s sF f W WT T
π= ≥ = ≥
Demo ExamplesDemo Examples1. 5 kHz analog bandwidth — sampled at 10, 5,
2.5, 1.25 kHz (notice the aliasing that arises when the sampling rate is below 10 kHz)
2. quantization to various levels — 12,9,4,2, and 1 bit quantization (notice the distortion introduced
h th b f bit i t l )
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when the number of bits is too low)3. music quantization — 14 bit audio quantized to
various levels:• 12 bit audio, 2 bit noise (white noise)• 10 bit audio, 4 bit noise (white noise)• 8 bit audio, 6 bit noise (colored)• 6 bit audio, 8 bit noise (signal correlated somewhat)
DiscreteDiscrete--Time (DT) Signals are SequencesTime (DT) Signals are Sequences
T
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• x[n] denotes the “sequence value at ‘time’ n”• Sources of sequences:
– Sampling a continuous-time signal x[n] = xc(nT) = xc(t)|t=nT
x n y n x n x nx n y n x n x nx n x n y n x n x n x n x n
↔ = + + +
↔ = + + +
+ ↔ = + + + + + +
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1 2[ ] [ ] Not a linear system!y n y n≠ + ⇒
Is system [ ] [ ] 2 [ 1] 3 time/shift invariant?y n x n x n= + + +
0 0 0
[ ] [ ] 2 [ 1] 3[ ] [ ] 2 [ 1] 3 System is time invariant!
y n x n x ny n n x n n x n n
= + + +− = − + − + + ⇒
Is system [ ] [ ] 2 [ 1] 3 causal?y n x n x n= + + +
[ ] depends on [ 1], a sample in the future System is not causal!y n x n +
⇒
Convolution ExampleConvolution Example1 0 3 1 0 3
[ ] [ ]0 otherwise 0 otherwise
What is [ ] for this system?
n nx n h n
y n
≤ ≤ ≤ ≤⎧ ⎧= =⎨ ⎨
⎩ ⎩
x[n],h[n]
n0 1 2 3 4 5
[ ] [ ] * [ ] [ ] [ ]y n x n h n h m x n m∞
∑
Solution :
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y[n]
n0 1 2 3 4 5 6 7
03
3
[ ] [ ] * [ ] [ ] [ ]
1 1 ( 1) 0 3
1 1 (7 ) 4 6
0 0, 7
m
n
m
m n
y n x n h n h m x n m
n n
n n
n n
=−∞
=
= −
= = −
⎧⋅ = + ≤ ≤⎪
⎪⎪
= ⋅ = − ≤ ≤⎨⎪⎪ ≤ ≥⎪⎩
∑
∑
∑
m0 1 2 3 4 5
m0 1 2 3 4 5
m0 1 2 3 4 5
n=0 n=1 n=2
n0 1 2 3 4 5
h[n]
n0 1 2 3 4 5
x[n]
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m0 1 2 3 4 5
m0 1 2 3 4 5
m0 1 2 3 4 5
m0 1 2 3 4 5 6
m0 1 2 3 4 5 6 7
m0 1 2 3 4 5 6 7 8
n=4n=3 n=5
n=6 n=7 n=8
Convolution ExampleConvolution ExampleThe impulse response of an LTI system is of the form: [ ] [ ] | | 1and the input to the system is of the form: [ ] [ ] | | 1,Determine the output of the system using the formulafor discrete
n
n
h n a u n a
x n b u n b b a
= <
= < ≠
convolution
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for discrete convolution.
0 0
1 1 1
SOLUTION:
[ ] [ ] [ ]
[ ] ( / ) [ ]
1 ( / ) [ ]1 ( / )
m n m
mn n
n m m n m
m m
n n nn
y n a u m b u n m
b a b u n b a b u n
a b b ab u na b b a
∞−
=−∞
−
= =
+ + +
= −
= =
⎡ ⎤ ⎡ ⎤− −= =⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦
∑
∑ ∑
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Convolution ExampleConvolution Example
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Convolution ExampleConvolution ExampleConsider a digital system with input [ ] 1 for
0,1,2,3 and 0 everywhere else, and with impulseresponse [ ] [ ], | | 1. Determine theresponse [ ] of this linear system.
n
x nn
h n a u n ay n
==
= <
SOLUTION:
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We recognize that [ ] can be written as the difference between two step functions, i.e., [ ] [ ] [ 4]. Hence we can solve for [ ] as the difference between the output of the linear system
x nx n u n u n y n= − −
1
1 1
1 1
with a step input and the output of the linear system with a delayed step input. Thus we solve for the response to a unit step as:
[ ] [ ] [ ] [ ]1
[ ] [ ] [ 4]
nn m
m
a ay n u m a u n m u na
y n y n y n
−∞−
−=−∞
⎡ ⎤−= − = ⎢ ⎥−⎣ ⎦
= − −
∑
Linear TimeLinear Time--Invariant SystemsInvariant Systems• easiest to understand• easiest to manipulate• powerful processing capabilities• characterized completely by their response to unit sample,
h(n), via convolution relationship∞ ∞
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*
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ], where denotes discrete convolution
∞ ∞
=−∞ =−∞
= ∗ = − = −
= ∗ ∗
∑ ∑k k
y n x n h n x k h n k h k x n k
y n h n x n
• basis for linear filtering
• used as models for speech production (source convolved with system)
h[n]x[n] y[n]=x[n] h[n]
Signal Processing OperationsSignal Processing Operations
341
is a delay of 1-sampleCan replace with delay element D
Issues with ‘Periodic’ SequencesIssues with ‘Periodic’ Sequences
( 1/ 2)Given a continuous time signal that is periodicwith period samples, is the sampledversion (sampling period ) a digital periodic sequence?
the DFT frequencies for an -point sequence
N TT
N
+
•
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/ )
' (2 / ( 1/ 2))
( 1/ 2)
are =(2 the signal periodicity frequencies are
the digital signal is not periodic with period
samples (in this case it actually is periodic with pe
k
k
N k
N k
N
ω π
ω π•
= +•
+(2 1)riod samples)N +
Periodic Sequences??Periodic Sequences??
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DFT for Finite Length DFT for Finite Length ggSequencesSequences
66
12
Finite Length SequencesFinite Length Sequences• consider a finite length (but not periodic)
sequence, x[n], that is zero outside the interval 0 1n N≤ ≤ −
1
( ) [ ]−
−= ∑N
nX z x n z
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0
2 / , 0,1, ..., 1
12 / 2 /( ) [ ] , 0,1,..., 10
( ) [ ]
evaluate at equally spaced points on the unit circle,
π
π π
=
= = −
− −= = −=
=
•
∑
∑n
X(z) Nj k Nz e k Nk
Nj k N j kn NX e x n e k Nn
X z x n z
--looks like DFT of periodic sequence!
Relation to Periodic SequenceRelation to Periodic Sequence[ ]
[ ],2 /
( ) [ ]
identical
-consider a periodic sequence, , consisting of an infinite
sequence of replicas of x[n]
- the Fourier coefficients, are then to the values
∞
=−∞
= +∑r
x n
X kj k N
x n x n r N0 2 4 6 8 10 12 14 16 18
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sample number
ampl
itude
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2 /( )of fπj k NX e
1 2 /[ ] [ ] , 0,1,..., 10
1 2 /[ ] [ ] , 0,1
a sequence
of length can be exactly represented by a DFT representation
or the finite duration sequence
of the form:
π
π
=>
− −= = −=
= =
∑
N
N j nk NX k x n e k Nn
j kn Nx n X k e nN
1,..., 1
0
−−
=∑
NN
k
Works for both finite sequence and for periodic
sequence
Periodic and Finite Length SequencesPeriodic and Finite Length Sequences
x[n]
n
X[k]
k
periodic signal => line spectrum in frequency
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x[n]
n
X(ejω)
ω
finite duration => continuous spectrum
in frequency
Sampling in Frequency (Time Domain Aliasing)Sampling in Frequency (Time Domain Aliasing)
1
0
[ ] 0 0 1
( ) [ ] 0 2
Consider a finite duration sequence: for
i.e., an point sequence, with discrete Fourier transform
Consider sampling the discrete Fourier transform bymul
ω ω ω π−
−
=
≠ ≤ ≤ −−
= ≤ ≤∑L
j j n
n
x n n LL
X e x n e
1
tiplying it by a signal that is defined as:−
∑N
j
( )jS e ω
0 1 2 3 4 N-1 kω=2πk/N
[ ]s n
. . .
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0
( ) [ 2 / ]
[ ] [ ]
( ) ( ) ( )
with time-domain representation
Thus we form the spectral sequence
which transforms
ω
ω ω ω
δ ω π
δ
=
∞
=−∞
= −
= −
= ⋅
∑
∑
j
k
r
j j j
S e k N
s n n rN
X e X e S e
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] ...
in the time domain to the convolution
δ∞ ∞
=−∞ =−∞
= ∗ = ∗ − = −
= + − + + +
∑ ∑r r
x n x n s n x n n rN x n rN
x n x n x n N x n N
-N 0 N 2N n
. . . . . .
Sampling in Frequency (Time Domain Aliasing)Sampling in Frequency (Time Domain Aliasing)
,1
If the duration of the finite duration signal satisfies the relation then only the first term in the infinite summation affects
the interval 0 and there is no time domain aliasing, i.e.,
N Ln L
≥≤ ≤ −
[ ] [ ] 0 1x n x n n L= ≤ ≤ −
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,If i.e., the number of frequency samples is smaller than the duration of the finite duration signal, then there is time domain aliasing and the resulting aliased signal (over the
N L<
0 1)
[ ] [ ] [ ] [ ] 0 1
interval satisfies the aliasing relation:
n L
x n x n x n N x n N n N
≤ ≤ −
= + + + − ≤ ≤ −
Time Domain Aliasing ExampleTime Domain Aliasing Example4
0
[ ] ( 1) [ ] [ ] 2 [ 1] 3 [ 2] 4 [ 3] 5 [ 4]
Consider the finite duration sequence
m
x n m n m n n n n nδ δ δ δ δ δ=
= + − = + − + − + − + −∑
-2 -1 0 1 2 3 4 5 6 7 8 n
72
[ ]The discrete Fourier transform of is computed and sampled at frequenciesaround the unit circle. The resulting sampled Fourier transform is inversetransformed back to the time domain. What is
x n N
[ ],0 1) 11, 5 4.
the resulting time domain signal, (over the interval for the cases and
x nn L N N N≤ ≤ − = = =
11 5, ) [ ] [ ]0 1. 4 0
[0] 5 [
SOLUTION:For the cases and we have no aliasing (since and we get over the interval For the case , the value is aliased, giving
(as opposed to 1 for
N N N L x n x nn L N n
x x
= = ≥ =≤ ≤ − = =
= 0]) with the remaining values unchanged.
13
DFT PropertiesDFT PropertiesPeriod=N Length=NSequence defined for all n Sequence defined for n=0,1,…,N-1DFT defined for all k DFT defined for all ω
Periodic Sequence Finite Sequence
h i DFT t ti ll b h if th
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[ ] [ ]
when using DFT representation, all sequences behave as if they
were infinitely periodic DFT is really the representation of the
extended periodic function,
alternative (equival
∞
=−∞
•
=>
= +
•
∑r
x n x n rN
[ ] [ ] [ ] ([ ])
ent) view is that all sequence indices must be interpreted modulo
modulo ∞
=−∞
= + = =∑ Nr
N
x n x n rN x n N x n
DFT Properties for Finite SequencesDFT Properties for Finite Sequences
• X[k], the DFT of the finite sequence x[n], can be viewed as a sampled version of the z-transform (or Fourier transform) of the finite sequence (used to design finite length filters via frequency sampling method)
74
• the DFT has properties very similar to those of the z-transform and the Fourier transform
• the N values of X[k] can be computed very efficiently (time proportional to N log N) using the set of FFT methods
• DFT used in computing spectral estimates, correlation functions, and in implementing digital filters via convolutional methods
Summary of DSPSummary of DSP--Part 1Part 1• speech signals are inherently bandlimited => must
sample appropriately in time and amplitude• LTI systems of most interest in speech processing; can
characterize them completely by impulse response, h(n)• the z-transform and Fourier transform representations
bl t ffi i tl i l i b th th ti
78
enable us to efficiently process signals in both the time and frequency domains
• both periodic and time-limited digital signals can be represented in terms of their Discrete Fourier transforms
• sampling in time leads to aliasing in frequency; sampling in frequency leads to aliasing in time => when processing time-limited signals, must be careful to sample in frequency at a sufficiently high rate to avoid time-aliasing
14
Digital FiltersDigital FiltersDigital FiltersDigital Filters
79
Digital FiltersDigital Filters• digital filter is a discrete-time linear, shift invariant
system with input-output relation:[ ] [ ] [ ] [ ] [ ]
( ) ( ) ( )( ) ( )
is the system function with ω
∞
=−∞
= ∗ = −
= ⋅
•
∑m
j
y n x n h n x m h n m
Y z X z H zH z H e
80
arg| (
( ) ( )
( ) ( ) ( )
( ) | ( ) |
is the system function with as the complex frequency response
ω ω ω
ω ω
•
= +
=j
j
j j jr i
j j j H e
H z H e
H e H e jH e
H e H e e )|
log ( ) log | ( ) | arg | ( ) |
log | ( ) | Re log ( )
arg | ( ) | Im log ( )
ω
ω ω ω
ω ω
ω ω
= +
⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
j j j
j j
j j
H e H e j H e
H e H e
j H e H e
real, imaginary representation
magnitude, phase representation
Digital FiltersDigital Filters• causal linear shift-invariant => h[n]=0 for n<0• stable system => every bounded input produces a
bounded output => a necessary and sufficient condition for stability and for the existence of
81
( )jH e ω
| [ ] |∞
=−∞
< ∞∑n
h n
Digital Filter ImplementationDigital Filter Implementation• input and output satisfy linear difference equation of the
form:
1 0[ ] [ ] [ ]
= =
− − = −∑ ∑N M
k rk r
y n a y n k b x n r
• evaluating z-transforms of both sides gives:
82
1 0
1 0
0
1
( ) ( ) ( )
( ) (1 ) ( )
( )( )( ) 1
− −
= =
− −
= =
−
=
−
=
− =
− =
= =−
∑ ∑
∑ ∑
∑
∑
N Mk r
k rk r
N Mk r
k rk r
Mr
rr
Nk
kk
Y z a z Y z b z X z
Y z a z X z b z
b zY zH zX z a z
canonic form showing poles
and zeros
Digital FiltersDigital Filters
• H(z) is a rational function inO
1(1 )−−∏M
A c z
1z−
83
XX
XX
O
O
O
1
1
1
(1 )( )
(1 )
=
−
=
−= =>
−
∏
∏
rr
N
kk
A c zH z M N
d z zeros, poles
1 1, 1
( )
• > <=>
| z | R R
H z
converges for with for stability
all poles of inside the unit circle for a stable, causal system
Ideal Filter ResponsesIdeal Filter Responses
84
15
FIR SystemsFIR Systems• if ak=0, all k, then
0 10
1
[ ] [ ] [ ] [ 1] ... [ ]
1. [ ] 00
otherwise
=
−
= − = + − + + − =>
= ≤ ≤=
∑M
r Mr
n
MM
y n b x n r b x n b x n b x n M
h n b n M
85
11
0 0
( ) (1 )
3. [ ] [ ]
2. zeros
if (symmetric, antisymmetric)ω
− −
= =
= => − =>
= ± −
∑ ∏MM
nn m
n m
j
H z b z c z M
h n h M nH(e / 2) ( ) ( ), real (symmetric), imaginary (anti-symmetric)ω ω ω−= =j j M jA e e A e
• linear phase filter => no signal dispersion because of non-linear phase => precise time alignment of events in signal
FIR Linear Phase Filter
event at t0 event at t0 + fixed delay
FIR FiltersFIR Filters• cost of linear phase filter designs
– can theoretically approximate any desired response to any degree of accuracy
• linear phase filters can be implemented with half the multiplications (because of the symmetry of the coefficients)
IIR SystemsIIR Systems
1 0[ ] [ ] [ ]
[ ] [ 1], [ 2],..., [ ][ ], [ 1],..., [ ]
depends on as well as
for
= =
= − + −
• − − −− −
• <
∑ ∑N M
k rk r
y n a y n k b x n r
y n y n y n y n Nx n x n x n M
M N
100
01
1
1
( )11
[ ] ( )
- partial fraction expansion
−
=−
− =
=
= =−−
=
∑∑
∑
Mr
r Nr k
Nk k k
kk
nk k
b zAH zd za z
h n A d u1
[ ]
[ ]
- for causal systems
is an infinite duration impulse response (see prob 2.9)=
∑N
kn
h n
IIR Design MethodsIIR Design Methods•• Impulse invariant transformation Impulse invariant transformation – match the
analog impulse response by sampling; resulting frequency response is aliased version of analog frequency response
•• Bilinear transformation Bilinear transformation – use a transformation to map an analog filter to a digital filter by warping the analog frequency scale (0 to infinity) to the digital frequency scale (0 to pi); use frequency pre-warping to preserve critical frequencies of transformation (i.e., filter cutoff frequencies) 101
IIR Filter DesignIIR Filter Design
102
18
Butterworth DesignButterworth Design
103
ChebyshevChebyshev Type I DesignType I Design
104
ChebyshevChebyshev BPF DesignBPF Design
105
ChebyshevChebyshev Type II DesignType II Design
106
Elliptic BPF DesignElliptic BPF Design
107
IIR FiltersIIR Filters• IIR filter issues:
– efficient implementations in terms of computations– can approximate any desired magnitude response
with arbitrarily small error– non-linear phase => time dispersion of waveform
108
• IIR design methods– Butterworth designs-maximally flat amplitude– Bessel designs-maximally flat group delay– Chebyshev designs-equi-ripple in either passband or
stopband– Elliptic designs-equi-ripple in both passband and
stopband
19
MatlabMatlab Elliptic Filter DesignElliptic Filter Design• use ellip to design elliptic filter
– [B,A]=ellip(N,Rp,Rs,Wn)– B=numerator polynomial—N+1 coefficients– A=denominator polynomial—N+1 coefficients– N=order of polynomial for both numerator and denominator– Rp=maximum in-band (passband) approximation error (dB)
109
p (p ) pp ( )– Rs=out-of-band (stopband) ripple (dB) – Wp=end of passband (normalized radian frequency)
• use filter to generate impulse response– y=filter(B,A,x)– y=filter impulse response– x=filter input (impulse)
• use zplane to generate pole-zero plot– zplane(B,A)
sec for 8kHz sampling rate sec for 10 kHz sampling rate sec for 16 kHz sampling rate sec for 20 kHz sampl
μμ
μμ
= − ∞ < < ∞= == == == =
ax n x nT nTTTT ing rate
The Sampling TheoremThe Sampling TheoremIf a signal xa(t) has a bandlimited Fourier transform Xa(jΩ)such that Xa(jΩ)=0 for Ω≥2πFN, then xa(t) can be uniquely reconstructed from equally spaced samples xa(nT), -∞<n<∞, if 1/T≥2 FN (FS≥2FN) (A-D or C/D converter)
xa(t)
116
a( )
xa(nT)
xa(nT) = xa(t) uT(nT), where uT(nT) is a periodic pulse train of period T, with periodic spectrum of period 2π/T
Decimation and InterpolationDecimation and Interpolation
Decimation, M=2 => T’=2T
122
Interpolation, L=2 => T’=T/2
DecimationDecimation[ ] ( ) ( ) 0, | | 2 ( )
1 2
1( ) ( ), | | ( )
( ) 0, 2 | | 2 ( )
Standard Sampling: begin with digitized signal:
a a N
s N
j Ta
j TN s N
x n x nT X j F a
F FT
X e X j bT T
X e F F F
π
π
π π
Ω
Ω
= ↔ Ω = Ω ≥
= ≥
= Ω Ω <
= ≤ Ω ≤ −
i
123
( ) can achieve perfect recovery of from digitized
ax tisamples under these conditions
DecimationDecimation
'
2[ ]
' 1/ ' 1( ) /
[ ] ( ' )
want to reduce sampling rate of sampled signal by factor of want to compute new signal
with sampling rate such that with no aliasing
d
s s
d a
Mx n
F T MT F M
x n x nT
≥
= = =
=
i
i
124
one solution is to dd a
i [ ] ( )[ ],
[ ] [ ]
ownsample by retaining one out of every samples of giving
a
d
x n x nTM x n
x n x nM
=
=
DecimationDecimation
125
' 2 2 ( )' 2 4 ( )
need to avoid aliasing for when we get aliasing for
s N
s N
F F M cF F M d
≥ =< =
ii
DecimationDecimation
1( 2 )/
0
[ ] [ ]1( ) ( )
DTFTs of and related by aliasing relationship:
i l tl (i t f l f )
d
Mj j k M
dk
x n x n
X e X eM
ω ω π−
−
=
= ∑
i
126
1' ( ' 2 )/
0
1( ) ( )
1' 2 ,
or equivalently (in terms of analog frequency):
assuming (i
Mj T j T k M
dk
s N
X e X eM
F FMT
π−
Ω Ω −
=
=
= ≥
∑
i
i
' '/1 1 1 1( ) ( ) ( ) ( )
1 ( ),' ' '
.e., no aliasing) we get:
j T j T M j Td a
a
X e X e X e X jM M M T
X jT T T
π π
Ω Ω Ω= = = Ω
= Ω − < Ω <
22
DecimationDecimation
[ ]/ (2 )
[ ]
to decimate by factor of with no aliasing, need toensure that the highest frequency in is no greaterthan thus we need to filter using an ideal lowpass
filter with response:
s
Mx n
F Mx n
i
i
1 | | / Mω <⎧
127
( jdH e ω 1 | | /
)0 / | |
using the appropriate lowpass filter, we can down-sample the reuslting lowpass-filtered signalby a factor of without aliasing
MM
M
ω ππ ω π
<⎧= ⎨ < ≤⎩
i
DecimationDecimation
' 1( ) ( ) ( ),' ' '
[ ],
using a lowpass filter gives:
if filter is used, the down-sampled signal, no longer represents the original analog signal,
j T j Td d a
d
W e H e X jT T T
w n
π πΩ Ω= Ω − < Ω <
i
i
128
( ), but instead the lowpass ax t( )
filtered version of
the combined operations of lowpass filteringand downsampling are called .
ax ti
decimation
InterpolationInterpolation[ ] ( )
( )'' / ,
[ ]
assume we have , (no aliasing) and we wish to increase the sampling rate by the integer factor of we need to compute a new sequence of samples of with period
i.e.,
a
a
i
x n x nTL
x tT T L
x n
=
=
i
i
( '') ( / ) It is clear that we can create the signal
a ax nT x nT L= =i
129
[ ] [ / ] 0, , 2 ,... for but we need to fill in the unknown samples by an interpolation process can readily show that what we wa
ix n x n L n L L= = ± ±
i
sin[ ( '' ) / ][ ] ( '') ( )[ ( '' ) / ]
'' / , [ ] ( )
sin[ ( ) / ][ ] ( '') ( )[ ( ) / ]
[ ] [ ]
nt is:
equivalently with we get
which relates to
i a ak
a
i a ak
i
nT kT Tx n x nT x kTnT kT T
T T L x n x nT
n k Lx n x nT x kn k L
x n x n
ππ
ππ
∞
=−∞
∞
=−∞
⎡ ⎤−= = ⎢ ⎥−⎣ ⎦
= =
⎡ ⎤−= = ⎢ ⎥−⎣ ⎦
∑
∑
i
i directly
InterpolationInterpolation[ / ] 0, , 2 ,...
[ ]0
[ ] 0, , 2 ,...,
implementing the previous equation by filtering the upsampled sequence
has the correct samples for but it has zero-valuedsamples in betwe
u
u
x n L n L Lx n
otherwisex n n L L
= ± ±⎧= ⎨
⎩= ± ±
i
i
[ ]
( ) ( )
en (from the upsampling operation) The Fourier transform of is simply:u
j j L
x n
X Xω ω
i
130
'' ''
''
( ) ( )
( ) ( ) ( )
( )2 /
Thus is periodic with two periods, namelywith period , due to
j j Lu
j T j T L j Tu
j Tu
X e X e
X e X e X e
X eL
ω ω
π
Ω Ω Ω
Ω
=
= =
i2upsampling) and due to being a digital signal π
InterpolationInterpolation
''
''
( ) ( )( ) ( )
2, '' / 2( )
(2 / ) ( ) | | 2( )
Plot of Plot of showing double
periodicity for DTFT of desired signal with
j T
j Tu
a Nj T
a X eb X e
L T Tc
T X j FX e
π
Ω
Ω
Ω
= =
Ω Ω ≤⎧= ⎨
131
( )0 2 | | / ''
( ) can obtain results of by applying
iN
X eF T
cπ π
= ⎨ < Ω ≤⎩i
2 / ,
(1/ '') ( ) 0 | | /( )
0 / | |
| | /( )
0 / | |
ideallowpass filter withgain (to restore amplitude)and cutoff frequency giving:N
j Lj
i
ji
LF T
T X e LX e
L
L LH e
L
ωω
ω
π π
ω ππ ω π
ω ππ ω π
=
⎧ ≤ <= ⎨
≤ ≤⎩<⎧
= ⎨ ≤ ≤⎩
InterpolationInterpolation[ ], ,
[ ]'' /
Original signal, at sampling period, is first upsampled to give signal withsampling period
lowpass filter removes images of original
u
x n Tx n
T T L=
i
i
132
[ ] ( '') ( / )
p g gspectrum giving:
i a ax n x nT x nT L= =
23
SR Conversion by NonSR Conversion by Non--Integer FactorsInteger Factors• T’=MT/L => convert rate by factor of M/L• need to interpolate by L, then decimate by M (why can’t
it be done in the reverse order?)
133
Lowpass Filter
Interpolation LPF
Decimation LPF
need to combine specifications of both LPFs and implement in a single stage of lowpass filtering
• can approximate almost any rate conversion with appropriate values of L and M
• for large values of L, or M, or both, can implement in stages, i.e., L=1024, use L1=32 followed by L2=32
Summary of DSPSummary of DSP--Part II Part II • digital filtering provides a convenient way of processing
signals in the time and frequency domains• can approximate arbitrary spectral characteristics via
either IIR or FIR filters, with various levels of approximation
134
• can realize digital filters with a variety of structures, including direct forms, serial and parallel forms
• once a digital signal has been obtain via appropriate sampling methods, its sampling rate can be changed digitally (either up or down) via appropriate filtering and decimation or interpolation
