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Unformatted text preview: Discretetime filters: convolution;
Fourier transform; lowpass and
highpass filters Input
→
x[n] Filter Output
→
y[n] n denotes the time variable: {…, 2, 1, 0, 1, 2, …}
x[n] denotes the sequence of input values:
{…, x[2], x[1], x[0], x[1], x[2], …}
y[n] denotes the sequence of output values:
{…, y[2], y[1], y[0], y[1], y[2], …}
Assume that
a) the principle of superposition holds ⇔ system is
linear, i.e. combining any two inputs in the form
Ax1[n] + Bx2[n]
results in an output of the form
Ay1[n] + By2[n]
2 b) the behavior of the system does not change with
time, i.e. a delayed version of any input
xd[n] = x[n  d]
produces an output with a corresponding delay
yd[n] = y[n – d]
Under these conditions, the system can be
characterized by its response, h[n], to a unit
impulse, δ[n], which is applied at time n = 0,
1 i.e. the particular input
Unit
x[n] = δ[n]
L
L Impulse
2 1 0 1 2 n
produces the output
y[n] = h[n] h[0]
h[1]
h[1]
h[2]
h[2] L
L
2 1 0 1 2 n Impulse
Response
3 The general input
∞
x[n] = ∑ x[k] δ[n – k]
k = ∞
will thus produce the output
∞
y[n] = ∑ x[k]h[nk]
k = ∞ Convolution sum 4 Discrete time Fourier transform
∞ X(ω) = ∑ x[n] eiωn
n = ∞ Inverse
x[n] = 1
2π π
⌠
X(ω) eiωn dω
⌡
π 5 Frequency Response
Suppose that we have the particular input
x[n] = eiωn
H(ω) eiωn
eiωn
What is the output?
→
→
y[n] = ∑ h[k] x [n  k]
k = eiωn ∑ h[k] eiωk
k
14243
14243
H(ω)
Frequency Response 6 Convolution Theorem
π
A general input
1⌠
x[n] = 2π ⌡ X(ω) eiωn dω
π will thus produce the output
π 1
y[n] = 2π ⌠ X(ω) H(ω) eiωn dω → Y(ω) = X(ω) H(ω)
⌡
 π 14243
Y(ω)
Convolution
Convolution of sequences x[n] and h[n] is denoted by
h[n] ∗ x[n] = ∑ x[k] h[n  k] = y[n] (say)
k 7 Matrix form:
OOO
Oh[0] h[1] h[2]
Oh[1] h[0] h[1] h[2]
h[2] h[1] h[0] h[1] h[2]
h[2] h[1] h[0] h[1]O
h[2] h[1] h[0] O
O
O
O
1444442444443
Toeplitz matrix M
x[2]
x[1]
x [0]
x [1]
x[2]
M = M
y[2]
y[1]
y[0]
y[1]
y[2]
M 8 Convolution is the result of multiplying polynomials:
(… + h[1]z + h[0] +h[1]z1 + …) (… + x[1]z + x[0] +
x[1]z1 + …) = (… + y[1]z + y[0] + y[1]z1 + …)
Example:
3 12
62
6 14 3
4
10
9 ↑↑↑
z5 z4 z3 3
2 1 5 2
2
4 1
1
5 2
20 8 0
4
00
17 13 2
↑
z2 01
5
4 2 x[n] 1 0
1 1 3 h[n] 2
2 13 14
6 ↑↑
z1 z0 y[n]
0 12
2 17 4
3
9 5
9 Discrete Time Filters (summary)
Discrete Time:
x[n]
→ h[n] y[n]
→ y[n] = ∑ x[k] h [nk]
k (Convolution) Discrete –time Fourier transform
X(ω) = ∑ x[n] eiωn
n Frequency domain representation
Y(ω) = H(ω) • X(ω) (Convolution theorem) 10
10 Toeplitz Matrix representation: M
y[2]
y[1]
y[0]
y[1]
y[2]
M = OOO
Oh[0] h[1] h[2]
Oh[1] h[0] h[1] h[2]
h[2] h[1] h[0] h[1] h[2]
h[2] h[1] h[0] h[1]O
h[2] h[1] h[0] O
O
O
O M
x[2]
x[1]
x [0]
x [1]
x[2]
M Filter is causal if y[n] does not depend on future
values of x[n].
Causal filters have h[n] = 0 for n < 0.
11
11 Filters
a) Lowpass filter example:
y[n] = ½ x[n] + ½ x [n1]
Filter representation:
y[n]
x[n]
→
h[n]
→ y[n] = ∑ x[k] h [nk]
k Impulse response is
½½ h[n]
L L
01 n
12
12 Frequency Response is
H(ω) = ∑ h[k]eiωk
k = ½ + ½ eiω
Rewrite as H(ω) = H(ω) eiφ(ω)
2
H(ω) = cos( ω /2) eiω//2 ; π ≤ ω ≤ π 13
13 b) Highpass Filter Example
y[n] = ½ x[n]  ½ x[n1]
Impulse response is
½ h[n]
L L
0 1 n ½ 14
14 Frequency response is
H(ω) = ½  ½ eiω
= i sin (ω/2) eiω/2
14243
14243 2
sin (ω/2) e–i(π//2 = 2
sin (ω/2) ei(π//2 + ω/2)  ω/2) ; π ≤ ω < 0
; 0<ω≤ π 15
15 ...
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 Spring '03
 GilbertStrang

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