23 April 2014
ECE 438 Essential Definitions and Relations

CT Metrics
4. x(t)y(t) 

1. Energy Ex 

2
x(t) dt
 x(t  )y()d



 x(t  kT )
1
2
x(t) dt
2. Power Px  lim

T  T
T /2
5. rep T [x(t)] 
3. root mean squared value xrms  Px
6. comb T [x(t)] 
T /2
k 

 x(kT) (t  kT)
k 

4. Area Ax 
 x(t)dt
Properties of the CT Impulse


T /2
1
x(t)dt
T  T 
T /2
5. Average value xavg  lim
1.
6. Magnitude Mx  max x(t)
2.
1. Energy Ex 


x[n]
2
n  
N
1
2
2. Power Px  lim
x[n]

N 
2N 1 n  N
3. root mean squared value xrms  Px
4. Area Ax 

 x[n]
and (t)  0, t  0

 t  
DT Metrics
 (t)dt  1

 (t  t0 ) x(t)dt  x(t 0 ),

provided x(t) is continuous at t  t 0
3. x(t) (t t 0 )  x(t 0 ) (t  t0 )
4. (t  t0 ) x(t)  x(t  t0 )
1
5.  (at  t 0 )  a  (t  t 0 / a)
n  
N
1
x[n]

N 
2N  1 n   N
5. Average xavg  lim
6. Magnitude Mx  max x[n]
 n  
CT Signals and Operators
1, | t|< 1 / 2
1. rect (t) = 
0, | t|> 1 / 2
CTFT
Sufficient conditions for existence

 |x(t )|dt  


 | x(t)|
2
or

Forward transform

X( f ) 
 x(t)e
 j 2 ft
dt

2. sinc (t) =
sin(t)
t
1
 t
3. (t )  lim rect 
 
 0 
Inverse transform

x(t) 
 X( f )e

j 2 ft
df
dt
23 April 2014
-2 -
Transform Relations
ECE 438 Essential Definitions ...
CT FT
1. rect (t)  sinc (f)
1. Linearity
CT FT
a1 x1(t)  a2 x 2 (t)  a1X1( f )  a2 X 2 ( f )
2. Scaling and shifting
t  t0 CT FT
 j2ft 0
x
 |a| X(af )e
 a 
j2 f 0 t
CT FT
CT FT
3. e j2 f 0 t  ( f  f 0 )
CT FT 1
4.
3. Modulation
x(t)e
CT FT
2. (t )  1 and 1  ( f )
cos(2f 0 t) 
2
{( f  f 0 )
 ( f  f 0 )}
CT FT
 X( f  f 0 )
DT Signals and Operators
4. Reciprocity
1, n  0
1. u[n] = 
0, n < 0
CT FT
X(t)  x( f )
5. Parseval

 |x(t )|
2

dt 

 | X( f )|
2
df
1,
2. [n] = 
0,

3. x[n]y[n] 
6. Initial Value

 x[n  k]y[k]
 x(t) dt


x(0) 
n0
k 

X(0) 
n0
 X( f ) df
x[n]
CT FT
x(t)y(t)  X( f )Y( f )
y[n]
4. Downsampling
y[n]  x[Dn]

7. Convolution
D
x[n]
D
y[n]
5. Upsampling
x[n / D], n / D integer - valued
y[n]  
else
 0,
8. Product
CT FT
x(t)y(t)  X ( f )Y( f )
9. Transform of periodic signal
CT FT 1
rep T [x(t)]  comb 1 [X( f )]
T
T
10. Transform of sampled signal
CT FT 1
comb T [x(t)]  rep 1 [X( f )]
T
T
Properties of the DT Impulse

1.
 [n]  1 and [n]  0, n  0
n 

2.
 [n  n0 ] x[n]  x[n0 ]
n 
3. [n  n0 ]x[n]  x[n  n0 ]
DTFT
Transform Pairs
Sufficient conditions for existence
23 April 2014


-3 -
ECE 438 Essential Definitions ...

 x[n] 2  
x[n]   or
n 
n 
Forward transform

x[n]
8. Downsampling
1 D1   2k 
Y ()   X
D k 0  D 
 x[n]e  jn
X() 
n 
y[n]
D
x[n]
y[n]
D
10. Upsampling
Y()  X(D)
Inverse transform

1
jn
x[n] 
X()e d

2 
Transform Pairs
DTFT
1. [n]  1
Transform Relations
DT FT
2. 1  2 rep2  [()]
1. Linearity
DT FT
a1 x1[n]  a2 x 2 [n]  a1X1 ()  a2 X 2 ()
DT FT
j n
3. e 0  2 rep 2  [(   0 )]
2. Shifting
DTFT
x[n  n0 ]  X( )e
 jn 0
3. Modulation
x[n]e
j0 n
DTFT
 X(   0 )
4. Parseval


1
2
 x[n]  2  X() d
n 

2
5. Initial Value
X(0) 

 x[n]
DT FT
4.
cos( 0 n) 
 rep 2  [(   0 )  (   0 )]
1, n  0,1,..., N 1 DTFT
x[n]  

else
0,
5.
sin( N / 2)  j ( N  1) / 2
X( ) 
e
sin(  / 2)
Relation between CT and DT
1. Time Domain
xd [n]  x a (nT)
n 

1
x[0] 
 X()d
2 
2. Frequency Domain
Xd ( d )  f s rep f s X a ( f a )
where f s 
6. Convolution
DT FT
x[n]y[n]  X()Y()
1
T
ZT
7. Product
DT FT

1
x[n]y[n] 
 X (  )Y ()d
2 
Forward transform

X(z) 
 x[n]z
n 
n
fa
d f s
2
,
23 April 2014
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Transform Relations
ECE 438 Essential Definitions ...
N1
X[k] 
1. Linearity
 j2  kn/ N
n0
ZT
a1 x1[n]  a2 x 2 [n] a1 X1 (z)  a2 X2 (z)
2. Shifting
ZT
x[n  n0 ] X( )z
 n0
3. Modulation
n
x[n]z0
 x[n]e
Inverse transform
1
x[n] 
N
N 1
 X[k]e
j2 kn /N
k 0
Transform Relations
ZT
 X(z / z0 )
1. Linearity
DFT
4. Multiplication by time index
ZT
d
n x[n]  z X(z)
dz
a1 x1[n]  a2 x 2 [n]  a1X1[k]  a2 X2 [k]
2. Shifting
DFT
x[n  n0 ]  X[k ]e
6. Convolution
ZT
x[n]y[n] X(z)Y(z)
7. Relation to DTFT
If X ZT(z) converges for |z| 1 , then
the DTFT of x[n] exists and
XDTFT( )  XZT(z) ze j
 j2 n 0k / N
3. Modulation
DFT
x[n]e j2nk0 / N  X[k  k0 ]
4. Reciprocity
DFT
X[n]  N x[k]
5. Parseval
Transform Pairs
N1

ZT
1.  [n]1, all z
ZT
2. a nu[n]
ZT
ZT
4. n a u[n]
ZT
5.  na u[n  1]
n
DFT
Forward transform
k0
n 0
x[0] 
2
2
N 1
1
1  az 1 
 X[k ]
X[0]   x[n]
1
1 , |z||a|
1  az
az
N 1
6. Initial Value
1
1 , |z||a|
1  az
3.  anu[n  1]
n
n 0
1
x[n] 
N
2
, | z|| a|
1 N 1
 X[k]
N k0
7. Periodicity
az 1
1 az 
1 2
, |z||a|
x[n  mN]  x[n] for all integers m
X[k  lN ]  X[k] for all integers l
8. Relation to DTFT of a finite length
sequence
Let x0 [n]  0 only for 0  n  N 1
23 April 2014
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ECE 438 Essential Definitions ...
DTFT
b
x0 [n]  X0 ( ) .
P{a  X  b}   f X (x)dx

Define x[n] 
a
 x [n  mN] ,

0
 f X (x)dx  1
m 
x[n]  X[k] ,

X[k]  X0 e
then
j2  k /N
f X (x)  0

DFT
2. Probability distribution function
FX (x)

.
x
 f X ( )d
FX (x) 
6. Periodic convolution

DFT
x[n] y[n]  X[k]Y[k]
f X (x) 
7. Product
DFT
x[n]y[n] 
1
X[k]  Y [k]
N
dFX (x)
dx
FX ()  0 FX () 1 FX (x) is a
non-decreasing function of x
Expectation Operator
Transform Pairs

E{g(X)} 
DFT
1.
 [n], 0  n  N  1 
1. Mean g(x)  x
1, 0  k  N  1
2. 2nd moment g(x)  x 2
DFT
2.
1, 0  n  N  1 
N [k], 0  k  N  1
DFT
3.
e j2 k 0 n /N , 0  n  N  1 
N [k - k 0 ], 0  k  N  1
DFT
cos(2  k0 n / N ), 0  n  N  1 
4.
N
 [k - k 0 ] +  [k - (N - k0 )],
2
0  k  N 1
j 0 n
DFT
sin (2k / N   0 ) / 2
4. Relation between mean, 2nd moment,
and variance
 2X  E{X2} (E{X})2
Two Random Variables
1. Bivariate probability density function
f XY (x, y)
bd
P{a  X  b,c  Y  d}    f XY (x,y)dxdy
ac
 
f XY (x, y)  0
  
,n  0,1,..., N  1 
sin (2k / N   0 )(N / 2)
2
3. Variance g(x)  (x  E{X})
  f XY (x,y)dxdy  1
5.
e
 g(x) f X (x)dx

e j( 2  k / N   0 )( N 1 ) /2
Random Signals
1. Probability density function f X (x)
2. Marginal probability density functions
f X (x) and f Y (y)

f X (x) 
 f XY (x, y)dy

23 April 2014
-6 
f Y (y) 

f XY (x, y)dx

3. Linearity of expectation
E{ag(X,Y)  bh(X,Y)}  aE{g(X,Y)} bE{h(X,Y)}
4. Correlation of X and Y
 
E{XY} 
  xyf
XY
(x, y )dxdy

5. Covariance of X and Y
 2XY  E{(X  E{X})(Y  E{Y})}
 E{XY}  E{X}E{Y}
6. Correlation coefficient of X and Y
XY 
 2XY
X Y
0 | XY |1
7. X and Y are independent if and only
if
f XY (x, y)  f X (x) f Y (y)
8. X and Y are uncorrelated if and only
if
E{XY}  E{X}E{Y}
Independence implies uncorrelatedness.
The converse is not true.
Discrete-time random signals
1. Autocorrelation function
r XX[m,m  n]  E{X[m]X[m  n]}
X[m] is wide-sense stationary if and only
if mean is constant, and autocorrelation
depends only on difference between
arguments, i.e.
E{X[n]}  X
r XX[n]  E{X[m]X[m  n]}
2. Cross-correlation function
r XY[m, m  n]  E{X[m]Y[m  n]}
ECE 438 Essential Definitions ...
X[m] and Y[m] 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.
r XY[n]  E{X[m]Y[m  n]}
3. Wide-sense stationary processes and
linear systems.
Let X[n] be wide-sense stationary, and
suppose that
Y[n]  h[n] X[n]   h[n  m]X[m] ,
m
then X[n] and Y[n] are jointly widesense stationary, and
Y   X  h[m]
m
rXY [n]  h[n]  rXX [n]
rYY [n]  h[n]  rXY [n]
Model for Speech
Our model for the speech waveform is
given by
s[n]  v[n]e[n] ,
where s[n] is the speech waveform, v[n]
is the vocal tract response that typically
contains one or more peaks
corresponding to resonant frequencies,
and e[n] is the excitation provided by
the glottis.
  [n  kP],
for a voiced phoneme,

e[n]   k

 white noise, for an unvoiced phoneme.
Here P is the pitch period in samples.
Short-Time Discrete Time Fourier
Transform (STDTFT)
Two interpretations:
23 April 2014
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1. DTFT of windowed waveform
centered at (fixed) n where w[n] is the
window function
S( , n)   s[k]w[n  k]e
ECE 438 Essential Definitions ...
that the baseband filter impulse response
h0 [n] satisfy
1 , n  0
.
h0 [n]   L
 0, n  kL
 j k
k
2. Response of narrowband filter
centered at (fixed) frequency  , after
shifting to base-band. The impulse
response of the narrowband filter is
h [n]  h0 [n]e j n . Here h0 [n] is the
impulse response of the base-band filter,
which is equivalent to the window
function w[n] in the first interpretation.
We thus have
S( , n)  h [n] s[n]e
 j n
The display of the 2-D image S( ,n) as
a function of  and n is called a
spectrogram. Let N be the length of the
window function w[n] ; and let P be the
pitch period. If N ? P , the spectrogram
is narrowband. If N  P , the
spectrogram is wideband.
There are no constraints on h0 [n] for
other values of n .
Miscellaneous
1. Geometric Series
N1
 zn 
n 0
2. N roots zk , k  0, , N 1 , of a
complex constant u0 , i.e. N solutions to
N
z u0  0
zk  u0
1/ N
e
cos(2 )  cos 2   sin 2 
Consider an L-channel modulated filter
bank with input x[n] and output from the
-th channel given by
cos 2  
sin  sin  
cos  cos  
hl [n]  h0 [n]e
 j 2 n / L
, l  0, ..., L  1 .
The output of the filter bank is formed
by summing the outputs of all L
channels
L 1
y[n]   yl [n]
l 0
A necessary and sufficient condition for
perfect reconstruction, i.e. y[n]  x[n] is
, k  0, ,N 1
sin(2 )  2 sin  cos 
sin 2  
where
j 2  k u 0 / N
3. Trigonometric Identities
sin(   )  sin  cos   cos  sin 
cos(   )  cos  cos  msin  sin 
Modulated Filter Bank
yl [n]  hl [n]  x[n] ,
1 z N
1 z
sin  cos  
cos  sin  
1 1
 cos(2 )
2 2
1 1
 cos(2 )
2 2
1
cos(   )  cos(   )
2
1
cos(   )  cos(   )
2
1
sin(   )  sin(   )
2
1
sin(   )  sin(   )
2