Convolution: Linear System Response - Prof. Leona Mcmillan, Study notes of Computer Science

Download Convolution: Linear System Response - Prof. Leona Mcmillan and more Study notes Computer Science in PDF only on Docsity! anne Convolution —aAlh. 7 DATA: BY THE NUMBERS Superposi'on
and
Shi2‐Invariance
 8/29/08
 Comp
665

–Convolu2on
 2
 € x[n] = ak xk [n] k ∑ Linear System →     y[n] = ak yk [n] k ∑ •  Recall
there
were
“basic”
signals
such
that:
 – Any
signal
can
be
represented
as
a
linear 
combina2on
of
these
building
block
signals
 – The
response
of
a
Linear
ShiK‐Invariant
system
to 
these
basic
signals
is
merely
a
scaled
sum
of
the 
system’s
response
to
the
building
block.
  •  Discrete systems:        unit input  •  Con1nuous systems:     unit impulse
 Convolu'on
Sums
 •  Convolu2on
Shorthand
 •  Or
alterna2vely…
 8/29/08
 Comp
665

–Convolu2on
 5
 € yo y1 y2  yn−1 yn                   = x0 h0 h1 h2  hn−1 hn                   + x1 h−1 h0 h1  hn−2 hn−1                   + x2 h−2 h−1 h0  hn−3 hn−2                   ++ xn−1 h−n +1 h−n +2 h−n +3  h0 h1                   + xn h−n h−n +1 h−n +2  h−1 h0                   = h0 h−1 h−2  h−n +1 h−n h1 h0 h−1  h−n +2 h−n +1 h2 h1 h0  h−n +3 h−n +2       hn−1 hn−2 hn−3  h0 h−1 hn hn−1 hn−2  h1 h0                   x0 x1 x2  xn−1 xn                   € y[n] = x[n]∗ h[n] = x[i ]h[n − i ] i =−∞ ∞ ∑ Example
 •  Assume
a
system
with
unit
response
 8/29/08
 Comp
665

–Convolu2on
 6
 0 0 1 2 1 1/3 0 1 2 3 4 1 1/2 ? h[n] x[n] δ[0] € x[n]∗ h[n] Example
cont.
 8/29/08
 Comp
665

–Convolu2on
 7
 € 1 3 1 2 5 6 2 3 5 6                 = 1 3 0 0 0 0 1 3 1 3 0 0 0 1 3 1 3 1 3 0 0 0 1 3 1 3 1 3 0 0 0 1 3 1 3 1 3                 1 1 2 1 1 2 1                 0 1 2 3 4 1 1/2 0 1 2 3 4 5/6 2/3 Unit
Impulse
 •  δΔ(t)
has
unit
area
 8/29/08
 Comp
665
–
Convolu2on
 10
 t Δ 1/Δ t kΔ x(kΔ) (k+1)Δ € = x(kΔ)δΔ (t − kΔ)Δ ⇓  x (t) = x(iΔ)δΔ (t − iΔ)Δ i =−∞ ∞ ∑ ⇓ limit as Δ → 0 x(t) = x(τ )δ(t − τ )dτ -∞ ∞ ∫ Signal represented as the composition of unit impulses Con'nuous
System
Response
 •  Con2nuous
Linear
ShiK‐Invariant
system
 8/29/08
 Comp
665
–
Convolu2on
 11
 Linear
 System
 δΔ(0) hΔ(0) €  x (t) = x(iΔ)δΔ (t − iΔ)Δ i =−∞ ∞ ∑ →  x (t) = x(iΔ)hΔ (t − iΔ)Δ i =−∞ ∞ ∑ ⇓ limit as Δ →∞ x(t) = x(τ )δΔ (t − τ )dτ -∞ ∞ ∫ → y(t) = x(τ )h(t − τ )dτ -∞ ∞ ∫ Convolution Integral Con'nuous
Convolu'on
 •  Nota2on
 •  Interpreta2on
 8/29/08
 Comp
665
–

Convolu2on
 12
 € y(t) = x(t) ∗ h(t) = x(τ )hΔ (t − τ )dτ -∞ ∞ ∫ € h(τ ) Flip →  h(−τ ) Shift →   h(t − τ ) Multiply →   x(τ )h(t − τ ) Integrate →    x(τ )h(t − τ )dτ −∞ ∞ ∫ Differen'ator
 •  Double
impulse,
“unit
doublet”
 •  Opera2onal
use
 8/29/08
 Comp
665
–

Convolu2on
 15
 t Δ 1/2Δ -Δ 1/2Δ € u1(t) € lim Δ → 0 € y(t) = x(t) ∗ u1(t) = dx(t) dt Why is it “unit”? Higher
Orders
 •  Unit
triplets
 •  And
beyond
 8/29/08
 Comp
665
–

Convolu2on
 16
 € u2(t) = u1(t) ∗ u1(t)→ x(t) ∗ u2(t) = d2x(t) dt 2 € un(t) = u1(t) ∗∗ u1(t) n times        → x(t) ∗ un(t) = dnx(t) dtn (n > 0) Discrete
Generaliza'on
 •  Differen2a2on,
two
variants
 •  Integra2on
 8/29/08
 Comp
665
–Convolu2on
 17
 € f1[n] = δ[n + 1] −δ[n] b1[n] = δ[n] −δ[n −1] 0 1 f1[n] -1 0 1 b1[n] € u[n] = δ[i ] i =0 ∞ ∑ -1 0 1 2 3 4 5 6 u[n] …