H
H: circulent matrix
h
h:
H
H's 0th column
H
H: DFT of
h
h;
H=FHh
H
F
H
h
contain equivalent info!
Effect of LSI system on an input
x
x
is easy to describe in the Fourier domain
(frequency domain)...
ie:
Yk=HkXk
Y
k
H
k
X
k
;
0≤k≤N−1
0
k
N
1
ie: pointwise multiplication
2 point smoother
Compute frequency response
H
H
if LSI system...
Hk=1N∑
n
=0N−1hne−(i2πNkn)=1N∑
n
=0N−112e−(i2πNkn)=1N(1+ei2πNk)=1N(e−(i2πNk2)+ei2πNk2)ei2πNk2=2NcosπNkeiπNk
H
k
1
N
n
N
1
0
h
n
2
N
k
n
1
N
n
N
1
0
1
2
2
N
k
n
1
N
1
2
N
k
1
N
2
N
k
2
2
N
k
2
2
N
k
2
2
N
N
k
N
k
(1)
Hk=2NcosπNk
H
k
2
N
N
k
(2)
where 0 is low frequency,
π
is high frequency,
2π
2
is low frequency, and
w
k
=2πnk
w
k
2
n
k
Therefore it is a lowpass filter
ie:
smoothing
≡
lowpass filtering
ie:
h
h
is a "square box".
The Frequency response is (using answer from test 1):
Hk=1NsinMπNksinπNke−(iπN(M−1)k)
H
k
1
N
M
N
k
N
k
N
M
1
k
(3)
Where
sinMπNksinπNk
M
N
k
N
k
is the "Dirichlet Kernel."
H0=
H
0
?
sin0=0
0
0
L'Hopitâl's rule to the rescue...
Hk
k
=0=dd
k
numerator

k
=0dd
k
denominator

k
=0=1N(−MπN)cosMπNk
k
=0(−πN)cosπNk
k
=0e0=1NM=MN
k
0
H
k
k
0
k
numerator
k
0
k
denominator
1
N
k
0
M
N
M
N
k
k
0
N
N
k
0
1
N
M
M
N
(4)
Hk=1N∑
n
=0N−1hne−(i2πNkn)=1N(1−e−(i2πNkn))=1N(ei2πNk2−e−(i2πNk2))e−(i2πNk2)=1N2isinπNke−(iπNk)
H
k
1
N
n
0
N
1
h
n
2
N
k
n
1
N
1
2
N
k
n
1
N
2
N
k
2
2
N
k
2
2
N
k
2
1
N
2
N
k
N
k
(5)
Hk=1N∑
n
=0N−1hne−(i2πNkn)=1Ne−(i2πNkm)
H
k
1
N
n
0
N
1
h
n
2
N
k
n
1
N
2
N
k
m
(6)
Hk=1N
H
k
1
N
,
∡
Hk=−(2πNkm)
∡
H
k
2
N
k
m
That is, delay
≡
"linear phase shift." (slope
=−(m2πN)
m
2
N