the Impulse Response of a Linear System

Auditory Perception:
2: Linear Systems
Signals en Systems:
To understand why the auditory system represents
sounds in the way it does, we need to cover some
elementary background of signal analysis:
- representation in the time domain (impulses)
- representation in the frequency domain (sines, Fourier)
and of linear systems analysis
Mathematical concepts for today:
- Dirac delta function (repeat)
- the superposition principle (linearity)
- the impulse response
- the relation between impulse response (time domain)
and transfer characteristic (frequency domain): Fourier
 for t  0
 (t)  
and
0 for t  0
The Dirac impulse:

 (t)dt  1

0

A simple approximation of the Dirac-impulse function
is a rectangular pulse, e.g. at time tk: δk
1
T
k


tk tk+∆T

(t  tk )  lim T 0 k
IN THE TIME DOMAIN:
EVERY SIGNAL CAN BE REPRESENTED BY A
SUM OF WEIGHTED DIRAC PULSES:
Example: A step signal is simply a sequence of pulses with height 1:
Step = sum of pulses (of width ∆t ) with weight 1
Mathematically: s(t) 

(t  t )
k
k 0
Any arbitrary signal can be approximated by a
sequence of pulses with variable height (weight):
k

x(tk )k

Sum of
weighted pulses:
x(t) 

 x(t )  
k
k
k
1
T
Approximation:
x(t) 

k
 x(t )  
k
k
k
tk tk+∆T
Exact:

x(t) 


 x( )  (t   )  d
 (t)
U(t)


0
This is the representation of signal x(t) in the
time domain by impulse functions.
Note: the STEP is the cumulative integral of the pulse: U(t) 
t
  ( )d

Linear Systems Theory:
the BLACK-BOX approach
‘known’
‘unknown’
‘known’
We restrict ourselves to
Linear Systems:
Our crucial concept for today: the Superposition Principle
LINEAIR
if input x1(t)
=>
Output y1(t)
and input x2(t)
=>
Output y2(t)
then it is required that:
 x1(t) +  x2(t) =>
 y1(t) +  y2(t)
N
In general:
a x
n
n1
n
(t) 
N
a y
n
n1
n
(t)
!!!!
N.B. This requirement holds for all t!
Our next important concept:
the Impulse Response of a Linear System:
Thanks to the Superposition Principle, we can compute
the response of a Linear System for an arbitrary input signal,
from the system’s Impulse Response!
How?
We just noted that each signal can be precisely
represented by a weighted sum of Dirac pulses:
x(t) 

 x( )  (t   )  d

(integration = very precise summation)
Now suppose that each Dirac impulse at time t-τ gives
an Impulse Response h(t-τ)

THEN: Precise prediction of the system’s response y(t)
is the same weighted sum of the Impulse Responses:
y(t) 

 x( )  h(t   )  d

We are dealing with causal systems only, meaning
that there will be a response ONLY AFTER there was
an input! (This means mathematically that h(t)=0 for t<0 )
y(t) 

 x( )  h(t   )  d

y(t) 

 x(t   )  h( )  d
0
time t  the present time
time   the PAST time
x(t   )  input signal  seconds ago
h( )  the STRENGTH of the PAST
the CONVOLUTION
INTEGRAL
How does
convolution
work?
y(t) 

 x(t   )  h( )  d
0
x(3) x(9)
x(t)  step at t 10



h(t)  exp(t /1.5)
h(0)
h(6)



y(9)  x(9) h(0)  x(8) h(1) 
00
000
 x(1) h(8)  x(0) h(9)
y(t) 

 x(t   )  h( )  d
0
y(10)  x(10) h(0)  x(9) h(1) 
 x(2) h(8)  x(1) h(9)
y(t) 

 x(t   )  h( )  d
0
y(11)  x(11) h(0)  x(10) h(1) 
 x(3) h(8)  x(2) h(9)
y(t) 

 x(t   )  h( )  d
0
y(12)  x(12) h(0)  x(11) h(1) 
 x(4) h(8)  x(3) h(9)
y(t) 

 x(t   )  h( )  d
0
y(15)  x(15) h(0)  x(14) h(1) 
 x(7) h(8)  x(6) h(9)
y(t) 

 x(t   )  h( )  d
0
y(20)  x(20) h(0)  x(19) h(1) 
 x(12) h(8)  x(11) h(9)
y(t) 

 x(t   )  h( )  d
0
The total convolution result:
HERE IS THE LINK WITH THE
FIRST LECTURE:
Convolution analysis of a Linear System is performed
in the TIME DOMAIN,
in which signals are expressed by the IMPULSE FUNCTION
and the associated IMPULSE RESPONSE
However, just like Signal Analysis, also Linear Systems
analysis can be performed in the FREQUENCY DOMAIN
In that case, signals are represented by sine waves, and the
system response is described by SINE-WAVE RESPONSES!
We recall (again) the SUPERPOSITION PRINCIPLE:
x(t)   pulses (FOURIER)
harmonic functions
y(t)  impulse responses (FOURIER)
(Convolution Integral)
y(t) 

 x(t   )  h( )  d
0
harmonic responses
(Transfer Characteristic)
Y()  H()  X()
Representation of the Linear System in the FREQUENCY DOMAIN:
The TRANSFER CHARACTERISTIC
H( )
Consists of:
• AMPLITUDE CHARACTERISTIC
• PHASE CHARACTERISTIC
G()
()

The Amplitude Characteristic describes how the amplitude
of each sine wave varies with frequency

t)
x(t)  sin(
The Phase Characteristic describes how the phase of each
sine wave varies with frequency
INPUT :
OUTPUT : y(t)  G()  sin( t  ( ))
Example: Transfer Characteristic of a Low-Pass Filter:
Model:
Frequency characteristic:
1/T
1/T
G( ) 
T
Impulse response
1
h( ) 
exp( /(RC))
RC
T
Step response
1
1 (T) 2
()  arctan( T)

s(t)  [1 exp(t /(RC))]

T  RC
TIME
CONSTANT

Y()  H(, E)  X()
Amplitude (dB)
EXAMPLE 2:
the PINNA acts as a
direction-dependent linear
acoustic filter
10
0
-10
H(, E)

Elevation-dependent
spectral shape cues
Elevation (deg)
+60
+40
(‘earprint’)
+20
0
-20
-40
Frequency (kHz)
End second lecture
End second lecture
End second lecture
End second lecture

Download Report