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EE 438 Essential Definitions and Relations
22 April 2020
ECE 438 Essential Definitions and Relations
CT Metrics
1. Energy
Ex = x(t)2
−∞
∞
∫ dt
E
x
=x(t)
2
-¥
¥
ò
dt
2. Power
Px = limT→∞1T
x(t) 2−T /2
T /2
∫ dt
P
x
=lim
T®¥
1
T
x(t)
2
-T/2
T/2
ò
dt
3. root mean squared value
xrms = Px
x
rms
=P
x
4. Area
Ax = x(t)−∞
∞
∫ dt
A
x
=x(t)
-¥
¥
ò
dt
5. Average value
xavg = limT→∞1T
x(t)−T /2
T /2
∫ dt
x
avg
=lim
T®¥
1
T
x(t)
-T/2
T/2
ò
dt
6. Magnitude
Mx = max−∞ < t < ∞
x(t)
M
x
=max
-¥
x(t)
DT Metrics
1. Energy
Ex = x[n]2
n =−∞
∞
∑
E
x
=x[n]
2
n=-¥
¥
å
2. Power
Px = limN→ ∞1
2N +1x[n]2
n =− N
N
∑
P
x
=lim
N®¥
1
2N+1
x[n]
2
n=-N
N
å
3. root mean squared value
xrms = Px
x
rms
=P
x
4. Area
Ax = x[n]n = −∞
∞
∑
A
x
=x[n]
n=-¥
¥
å
5. Average
xavg = limN→ ∞1
2N + 1x[n]
n = − N
N
∑
x
avg
=lim
N®¥
1
2N+1
x[n]
n=-N
N
å
6. Magnitude
Mx = max−∞
x[n]
M
x
=max
-¥
x[n]
CT Signals and Operators
1.
rect (t) =1, | t|< 1 / 20, | t|> 1 / 2⎧ ⎨ ⎩
rect(t)=
1,|t|<1/2
0,|t|>1/2
ì
í
î
2.
sinc(t) = sin(πt)πt
sinc(t)=
sin(pt)
pt
3.
δ(t ) = limΔ→0
1Δrect t
Δ⎛ ⎝
⎞ ⎠
d(t)=lim
D®0
1
D
rect
t
D
æ
è
ö
ø
4.
x(t)∗y(t) = x(t − τ)y(−∞
∞
∫ τ)dτ
x(t)*y(t)=x(t-t)y(
-¥
¥
ò
t)dt
5.
repT[x(t)] = x(t − kT )k =−∞
∞
∑
rep
T
[x(t)]=x(t-kT)
k=-¥
¥
å
6.
combT[x(t)] = x(kT)k =−∞
∞
∑ δ(t − kT)
comb
T
[x(t)]=x(kT)
k=-¥
¥
å
d(t-kT)
Properties of the CT Impulse
1.
δ(t)− ∞
∞
∫ dt = 1 and δ(t) = 0, t ≠ 0
d(t)
-¥
¥
ò
dt=1 and d(t)=0, t¹0
2.
δ(t − t0)− ∞
∞
∫ x(t)dt = x(t0), provided x(t) is continuous at t = t0
d(t-t
0
)
-¥
¥
ò
x(t)dt=x(t
0
),
provided x(t) is continuous at t=t
0
3.
x(t)δ( t − t0 ) ≡ x(t 0)δ(t − t0 )
x(t)d(t-t
0
)ºx(t
0
)d(t-t
0
)
4.
δ(t − t0 )∗ x(t) = x(t − t0 )
d(t-t
0
)*x(t)=x(t-t
0
)
5.
δ(at − t0) ≡1aδ(t − t0 / a)
d(at-t
0
)º
1
a
d(t-t
0
/a)
CTFT
Sufficient conditions for existence
|x(t )− ∞
∞
∫ |dt < ∞ or | x(t)−∞
∞
∫ |2 dt
|x(t)
-¥
¥
ò
|dt<¥ or |x(t)
-¥
¥
ò
|
2
dt
Forward transform
X( f ) = x(t)e− j 2πft−∞
∞
∫ dt
X(f)=x(t)e
-j2pft
-¥
¥
ò
dt
Inverse transform
x(t) = X( f )ej 2π ft−∞
∞
∫ df
x(t)=X(f)e
j2pft
-¥
¥
ò
df
Transform Relations
1. Linearity
a1x1( t) + a2x2(t) ↔CTFT
a1X1( f ) + a2X2( f )
a
1
x
1
(t)+a
2
x
2
(t)«
CTFT
a
1
X
1
(f)+a
2
X
2
(f)
2. Scaling and shifting
x t − t0a
⎛ ⎝
⎞ ⎠ ↔CTFT
|a|X(af )e− j2π ft 0
x
t-t
0
a
æ
è
ö
ø
«
CTFT
|a|X(af)e
-j2pft
0
3. Modulation
x(t)e j2πf 0t ↔CTFT
X( f − f 0)
x(t)e
j2pf
0
t
«
CTFT
X(f-f
0
)
4. Reciprocity
X(t) ↔CTFT
x(− f )
X(t)«
CTFT
x(-f)
5. Parseval
|x(t )− ∞
∞
∫ |2 dt = |X( f )−∞
∞
∫ |2 df
|x(t)
-¥
¥
ò
|
2
dt=|X(f)
-¥
¥
ò
|
2
df
6. Initial Value
X(0) = x(t)−∞
∞
∫ dt
x(0) = X( f )−∞
∞
∫ df
X(0)=x(t)
-¥
¥
ò
dt
x(0)=X(f)
-¥
¥
ò
df
7. Convolution
x(t)∗y(t) ↔CTFT
X( f )Y( f )
x(t)*y(t)«
CTFT
X(f)Y(f)
8. Product
x(t)y(t) ↔CTFT
X ( f )∗Y( f )
x(t)y(t)«
CTFT
X(f)*Y(f)
9. Transform of periodic signal
repT[x(t)] ↔CTFT 1
Tcomb 1
T
[X( f )]
rep
T
[x(t)]«
CTFT
1
T
comb
1
T
[X(f)]
10. Transform of sampled signal
combT[x(t)] ↔CTFT 1
Trep 1
T
[X( f )]
comb
T
[x(t)]«
CTFT
1
T
rep
1
T
[X(f)]
Transform Pairs
1.
rect (t) ↔CTFT
sinc(f)
rect(t)«
CTFT
sinc(f)
2.
δ(t ) ↔CTFT
1 and 1 ↔CTFT
δ( f )
d(t)«
CTFT
1 and 1«
CTFT
d(f)
3.
e j2πf 0t ↔CTFT
δ( f − f 0)
e
j2pf
0
t
«
CTFT
d(f-f
0
)
4.
cos(2πf0t) ↔CTFT 1
2{δ( f − f 0 )
+ δ( f + f 0)}
cos(2pf
0
t)«
CTFT
1
2
{d(f-f
0
)
+d(f+f
0
)}
DT Signals and Operators
1.
u[n] =1, n ≥ 00, n < 0⎧ ⎨ ⎩
u[n]=
1,n³0
0,n<0
ì
í
î
2.
δ[n] =1, n = 00, n ≠ 0⎧ ⎨ ⎩
d[n]=
1,n=0
0,n¹0
ì
í
î
3.
x[n]∗y[n] = x[n − k]y[k]k =−∞
∞
∑
x[n]*y[n]=x[n-k]y[k]
k=-¥
¥
å
4. Downsampling
Dx[n] y[n]
D
x[n] y[n]
y[n] = x[Dn]
y[n]=x[Dn]
5. Upsampling
Dx[n] y[n]
D
x[n] y[n]
y[n] =x[n / D], n / D integer - valued
0, else⎧ ⎨ ⎩
y[n]=
x[n/D],n/D integer-valued
0, else
ì
í
î
Properties of the DT Impulse
1.
δ[n]n=−∞
∞
∑ = 1 and δ[n] = 0, n ≠ 0
d[n]
n=-¥
¥
å
=1 and d[n]=0,n¹0
2.
δ[n − n0]n=−∞
∞
∑ x[n] = x[n0 ]
d[n-n
0
]
n=-¥
¥
å
x[n]=x[n
0
]
3.
δ[n − n0 ]∗x[n] = x[n − n0]
d[n-n
0
]*x[n]=x[n-n
0
]
DTFT
Sufficient conditions for existence
x[n]n=−∞
∞
∑ <∞ or x[n]2n=−∞
∞
∑ < ∞
x[n]
n=-¥
¥
å
<¥ or x[n]
2
n=-¥
¥
å
<¥
Forward transform
X(ω) = x[n]e− jωnn= −∞
∞
∑
X(w)=x[n]e
-jwn
n=-¥
¥
å
Inverse transform
x[n] = 12π
X(ω)e jωnd−π
π
∫ ω
x[n]=
1
2p
X(w)e
jwn
d
-p
p
ò
w
Transform Relations
1. Linearity
a1x1[n] + a2x2[n] ↔DTFT
a1X1(ω) + a2X2(ω)
a
1
x
1
[n]+a
2
x
2
[n]«
DTFT
a
1
X
1
(w)+a
2
X
2
(w)
2. Shifting
x[n − n0 ]↔DTFT
X(ω )e− jωn0
x[n-n
0
]«
DTFT
X(w)e
-jwn
0
3. Modulation
x[n]e jω0 n ↔DTFT
X(ω −ω 0)
x[n]e
jw
0
n
«
DTFT
X(w-w
0
)
4. Parseval
x[n]2n=−∞
∞
∑ = 12π X(ω)2d
−π
π
∫ ω
x[n]
2
n=-¥
¥
å
=
1
2p
X(w)
2
d
-p
p
ò
w
5. Initial Value
X(0) = x[n] n=−∞
∞
∑
x[0] = 12π
X(ω)d−π
π
∫ ω
X(0)=x[n]
n=-¥
¥
å
x[0]=
1
2p
X(w)d
-p
p
ò
w
6. Convolution
x[n]∗y[n] ↔DTFT
X(ω)Y (ω)
x[n]*y[n]«
DTFT
X(w)Y(w)
7. Product
x[n]y[n] ↔DTFT 1
2πX (ω − µ)Y(µ)d
−π
π
∫ µ
x[n]y[n]«
DTFT
1
2p
X(w-m)Y(m)d
-p
p
ò
m
8. Downsampling
Dx[n] y[n]
D
x[n] y[n]
Y (ω) = 1D
X ω − 2πkD
⎛ ⎝
⎞ ⎠
k= 0
D−1
∑
Y(w)=
1
D
X
w-2pk
D
æ
è
ö
ø
k=0
D-1
å
9. Upsampling
Dx[n] y[n]
D
x[n] y[n]
Y (ω) = X(Dω)
Y(w)=X(Dw)
Transform Pairs
1.
δ[n] ↔DTFT
1
d[n]«
DTFT
1
2.
1 ↔DTFT
2π rep2π[δ(ω)]
1«
DTFT
2prep
2p
[d(w)]
3.
e jω 0n ↔DTFT
2π rep2π[δ(ω − ω0 )]
e
jw
0
n
«
DTFT
2prep
2p
[d(w-w
0
)]
4.
cos(ω0n) ↔DTFT
π rep2π[δ(ω − ω0 ) + δ(ω +ω0 )]
cos(w
0
n)«
DTFT
prep
2p
[d(w-w
0
)+d(w+w
0
)]
5.
x[n]= 1, n = 0,1,...,N −10, else
⎧⎨⎪
⎩⎪↔DTFT
X(ω ) = psincN ω( )e− jω (N−1)/2
where psincN ω( ) !sin(ωN / 2)sin(ω / 2)
Relation between CT and DT
1. Time Domain
xd[n] = xa (nT)
x
d
[n]=x
a
(nT)
2. Frequency Domain
Xd (ωd) = fsrep f s Xa ( f a )[ ] f a =ωd f s2π,
where fs =1T
X
d
(w
d
)=f
s
rep
f
s
X
a
(f
a
)
[]
f
a
=
w
d
f
s
2p
,
where f
s
=
1
T
ZT
Forward transform
Transform Relations
1. Linearity
2. Shifting
x[n− n0]↔ZT
X (z)z−n0
x[n-n
0
]«
ZT
X(z)z
-n
0
3. Modulation
4. Multiplication by time index
5. Convolution
6. Relation to DTFT
If
converges for
, then the DTFT of
exists and
Transform Pairs
1.
2.
3.
4.
5.
DFT
Forward transform
Inverse transform
Transform Relations
1. Linearity
2. Shifting
3. Modulation
4. Reciprocity
5. Parseval
6. Initial Value
7. Periodicity
8. Relation to DTFT of a finite length sequence
Let
only for
.
Define
,
,
then
.
9. Periodic convolution
x[n]⊗ y[n]↔DFT
X[k]Y [k]
x[n]Äy[n]«
DFT
X[k]Y[k]
10. Product
x[n]y[n]↔DFT 1
NX[k]⊗Y [k]
x[n]y[n]«
DFT
1
N
X[k]ÄY[k]
Transform Pairs
1.
2.
3.
4.
5.
e jω0n ,n = 0,1,...,N −1 ↔DFT
psincN 2πk / N −ω 0( )e− j (2πk /N−ω0 )(N−1)/2
e
jw
0
n
,n=0,1,...,N-1«
DFT
psinc
N
2pk/N-w
0
()
e
-j(2pk/N-w
0
)(N-1)/2
Random Signals
1. Probability density function
2. Probability distribution function
is a non-decreasing function of
Expectation Operator
1. Mean
2. 2nd moment
3. Variance
4. Relation between mean, 2nd moment, and variance
Two Random Variables
1. Bivariate probability density function
2. Marginal probability density functions
and
3. Linearity of expectation
4. Correlation of
and
E{XY} = xyfXY (x, y−∞
∞
∫−∞
∞
∫ )dxdy
E{XY}=xyf
XY
(x,y
-¥
¥
ò
-¥
¥
ò
)dxdy
5. Covariance of
and
6. Correlation coefficient of
and
7.
and
are independent if and only if
8.
and
are uncorrelated if and only if
Independence implies uncorrelatedness. The converse is not true.
Discrete-time random signals
1. Autocorrelation function
is wide-sense stationary if and only if mean is constant, and autocorrelation depends only on difference between arguments, i.e.
2. Cross-correlation function
and
are jointly wide-sense stationary if and only if they are individually wide-sense stationary and their cross-correlation depends only on difference between arguments, i.e.
3. Wide-sense stationary processes and linear systems.
Let
be wide-sense stationary, and suppose that
,
then
and
are jointly wide-sense stationary, and
µY = µX h[m]m∑
rXY [n] = h[n]∗ rXX[n]rYY [n] = h[n]∗ rXY [−n]
m
Y
=m
X
h[m]
m
å
r
XY
[n]=h[n]*r
XX
[n]
r
YY
[n]=h[n]*r
XY
[-n]
Model for Speech
Our model for the speech waveform is given by
s[n] = v[n]∗e[n]
s[n]=v[n]*e[n]
,
where
s[n]
s[n]
is the speech waveform,
v[n]
v[n]
is the vocal tract response that typically contains one or more peaks corresponding to resonant frequencies, and
e[n]
e[n]
is the excitation provided by the glottis.
e[n]=δ [n − kP],
k∑ for a voiced phoneme,white noise, for an unvoiced phoneme.
⎧⎨⎪
⎩⎪
e[n]=
d[n-kP],
k
å
for a voiced phoneme,
white noise,for an unvoiced phoneme.
ì
í
ï
î
ï
Here
P
P
is the pitch period in samples.
Short-Time Discrete Time Fourier Transform (STDTFT)
Two interpretations:
1. DTFT of windowed waveform centered at (fixed)
n
n
where
w[n]
w[n]
is the window function
!S(ω ,n)= s[k]w[n−k]e− jωk
k∑
2. Response of narrowband filter centered at (fixed) frequency
ω
w
, after shifting to base-band. The impulse response of the narrowband filter is
hω [n] = h0[n]e− jωn
h
w
[n]=h
0
[n]e
-jwn
. Here
h0[n]
h
0
[n]
is the impulse response of the base-band filter, which is equivalent to the window function
w[n]
w[n]
in the first interpretation. We thus have
!S(ω ,n)= hω[n]∗s[n]( )e+ jωn
The display of the 2-D image
!S(ω ,n)
as a function of
ω
w
and
n
n
is called a spectrogram. Let
N
N
be the length of the window function
w[n]
w[n]
; and let
P
P
be the pitch period. If
N ≫ P
N≫P
, the spectrogram is narrowband. If
N ≈ P
N»P
, the spectrogram is wideband.
Modulated Filter Bank
Consider an L-channel modulated filter bank with input
x[n]
x[n]
and output from the
ℓ
ℓ
-th channel given by
yℓ[n] = hℓ[n]∗ x[n]
y
ℓ
[n]=h
ℓ
[n]*x[n]
,
where
hℓ[n] = h0[n]e− j2πn /L , ℓ = 0, ..., L −1
h
ℓ
[n]=h
0
[n]e
-j2pn/L
,ℓ=0,...,L-1
.
The output of the filter bank is formed by summing the outputs of all L channels
y[n] = yℓ[n]
ℓ=0
L−1
∑
y[n]=y
ℓ
[n]
ℓ=0
L-1
å
A necessary and sufficient condition for perfect reconstruction, i.e.
y[n] ≡ x[n]
y[n]ºx[n]
is that the baseband filter impulse response
h0[n]
h
0
[n]
satisfy
h0[n] =1L , n = 00, n = kL
⎧⎨⎩
h
0
[n]=
1
L
,n=0
0,n=kL
ì
í
î
.
There are no constraints on
h0[n]
h
0
[n]
for other values of
n
n
.
Miscellaneous
1. Geometric Series
znn= 0
N−1
∑ = 1− zN
1− z
z
n
n=0
N-1
å
=
1-z
N
1-z
2.
N
N
roots
zk , k = 0,…,N −1
z
k
,k=0,…,N-1
, of a complex constant
u0
u
0
, i.e.
N
N
solutions to
zN −u0 = 0
z
N
-u
0
=0
zk = u01 / N e j 2 πk+ ∠u 0( ) / N , k = 0,…,N −1
z
k
=u
0
1/N
e
j2pk+Ðu
0
()
/N
,k=0,…,N-1
3. Trigonometric Identities
sin(α ± β) = sinα cosβ ± cosα sinβcos(α ± β) = cosα cosβ ∓ sinα sinβsin(2α ) = 2sinα cosαcos(2α ) = cos2α − sin2α
sin2α = 12−12cos(2α )
cos2α = 12+12cos(2α )
sinα sinβ = 12cos(α − β) − cos(α + β)[ ]
cosα cosβ = 12cos(α − β) + cos(α + β)[ ]
sinα cosβ = 12sin(α + β) + sin(α − β)[ ]
cosα sinβ = 12sin(α + β) − sin(α − β)[ ]
sin(a±b)=sinacosb±cosasinb
cos(a±b)=cosacosb∓sinasinb
sin(2a)=2sinacosa
cos(2a)=cos
2
a-sin
2
a
sin
2
a=
1
2
-
1
2
cos(2a)
cos
2
a=
1
2
+
1
2
cos(2a)
sinasinb=
1
2
cos(a-b)-cos(a+b)
[ ]
cosacosb=
1
2
cos(a-b)+cos(a+b)
[ ]
sinacosb=
1
2
sin(a+b)+sin(a-b)
[ ]
cosasinb=
1
2
sin(a+b)-sin(a-b)
[ ]