ADSP-Lecture 2
LTI System, Convolution, Difference Equations
Classification of Discrete-time Systems
Memoryless Systems: If the output of the system at an instant 𝑛 only depends on the
input sample at that time (and not on past or future samples) then the system is
called memoryless or static,
e.g. 𝑦(𝑛) = 𝑎𝑥(𝑛) + 𝑏𝑥2(𝑛)
Otherwise, the system is said to be dynamic or to have memory,
e.g. 𝑦(𝑛) = 𝑥(𝑛) − 4𝑥(𝑛 − 2)
Causal vs. Non-causal Systems
In a causal system, the output at any time n only depends on the present and past inputs.
An example of a causal system:
𝑦(𝑛) = 𝐹[𝑥(𝑛), 𝑥(𝑛 − 1), 𝑥(𝑛 − 2), … ]
y[n] = x[n] − x[n−1] (Forward Difference)
All other systems are non-causal.
A subset of non-causal system where the system output, at any time n only depends on
future inputs is called anti-causal.
𝑦(𝑛) = 𝐹[𝑥(𝑛 + 1), 𝑥(𝑛 + 2), … ]
y[n] = x[n+1] − x[n] (Backward Difference)
Stable vs. Unstable Systems
Unstable systems exhibit erratic and extreme behavior. BIBO stable systems are those
producing a bounded output for every bounded input:
x ( n)  M x    y ( n)  M y  
Example:
Stable or unstable?
y (n) = y 2 (n − 1) + x(n)
x( n) = C ( n) Bounded signal
Solution:
y (0) = C , y (1) = C 2 , y (2) = C 4 ,..., y (n) = C 2 n
1 C  
unstable
Linear vs. Non-linear Systems
Superposition principle: 𝑇[𝑎𝑥1(𝑛) + 𝑏𝑥2(𝑛)] = 𝑎𝑇[𝑥1(𝑛)] + 𝑏𝑇[𝑥2 (𝑛)]
A relaxed linear system with zero input produces a zero output.
Additivity property
Scaling property
Linear vs. Non-linear Systems
Example:
y ( n) = x ( n 2 )
Solution:
y1 (n) = x1 (n 2 )
Linear or non-linear?
y2 (n) = x2 (n 2 )
y3 (n) = T (a1 x1 (n) + a2 x2 (n)) = a1x1 (n 2 ) + a2 x2 (n 2 )
a1 y1 (n) + a2 y2 (n) = a1 x1 (n 2 ) + a2 x2 (n 2 )
Example:
Linear!
y (n) = e x ( n )
Useful Hint: In a linear system, zero input results in a zero
output!
x( n) = 0  y ( n) = 1
Non-linear!
System Examples (continue)
Nonlinear System.
Eg. w[n] = log10(|x[n]|) is not linear.
Time-invariant System:
If y[n] = T{x[n]}, then y[n−n0] = T{x[n −n0]}
The accumulator is a time-invariant system.
The compressor system (not time-invariant)
y[n] = x[Mn], − < n < .
Time-invariant vs. Time-variant Systems
If input-output characteristics of a system do not change with time then it is called
time-invariant or shift-invariant. This means that for every input 𝑥(𝑛) and every shift
𝑘
T
T
x( n) ⎯
⎯→
y ( n)  x ( n − k ) ⎯
⎯→
y (n − k )
Time-invariant vs. Time-variant Systems
Time-invariant example: differentiator
T
x( n) ⎯
⎯→
y ( n) = x( n) − x( n − 1)
T
x( n − 1) ⎯
⎯→
y (n − 1) = x( n − 1) − x(n − 2)
T
x ( n) ⎯
⎯→
y (n) = x(n).Cos (0 .n)
Time-variant example: modulator
x(n -1) ¾T¾
® x(n -1).Cos(w0 .n) ¹ y(n -1)
y(n -1) = x(n -1).Cos(w0.(n -1))
Example:
Accumulator System
Is the system memory-less?
Is the system causal?
Is the system Linear?
Is the system Time-invariant?
Is the system stable?
yn =
n
 xk 
k = −
Linear Time-Invariant (LTI) Systems
LTI systems have two important characteristics:
Time invariance: A system 𝑇 is called time-invariant or shift-invariant if input-output
characteristics of the system do not change with time
T
T
x ( n) ⎯
⎯→
y ( n)  x ( n − k ) ⎯
⎯→
y (n − k )
Linearity: A system 𝑇 is called linear iff
𝑇[𝑎𝑥1(𝑛) + 𝑏𝑥2(𝑛)] = 𝑎𝑇[𝑥1(𝑛)] + 𝑏𝑇[𝑥2 (𝑛)]
Why do we care about LTI systems?
Availability of a large collection of mathematical techniques
Many practical systems are either LTI or can be approximated by LTI systems.
Linear and Time Invariant Example
Linear Time Invariant Systems
A system that is both linear and time invariant is called a linear time invariant (LTI)
system.
By setting the input x[n] as [n], the impulse function, the output h[n] of an LTI system
is called the impulse response of this system.
Time invariant: when the input is [n-k], the output is h[n-k].
Remember that the x[n] can be represented as a linear combination of delayed
impulses

xn =
 xk  n − k 
k = −
Impulse Response
Impulse Response of LTI Systems
ℎ(𝑛): the response of the LTI system to the input unit sample (𝑛), i.e. ℎ(𝑛) = 𝑇{𝑥(𝑛)}

y(n) = T[x(n)] = x(k )h(n − k ) = x(n) * h(n)
k = −
An LTI system is completely characterized by a single impulse response ℎ(𝑛).
Response of the system to the input
Convolution
unit sample sequence at n=k
sum
Impulse Response Example
The impulse response of the system
y[n] = 1x[n] +  2 x[n − 1] + 3 x[n − 2] +  4 x[n − 3]
is obtained by setting 𝑥[𝑛] = 𝛿[𝑛] resulting in
h[n] = 1 [n] +  2 [n − 1] + 3 [n − 2] +  4 [n − 3]
The impulse response is thus a finite-length sequence of length 4 given by
{h[n]} = {1,  2 , 3 ,  4}

Impulse Response Example
Example - The impulse response of the discrete-time accumulator
n
y[n] =
 x[]
 = −
is obtained by setting 𝑥[𝑛] = 𝛿[𝑛] resulting in
h[n] =
n
  [] = [n]
 = −
FIR and IIR systems
Linear Time Invariant Systems (continue)
• Hence







xk hn − k 
yn = T   xk  n − k  =  xk T  n − k  =


k = −
k = −
 k = −

• Therefore, a LTI system is completely characterized
by its impulse response h[n].
Stable and Causal LTI Systems
An LTI system is (BIBO) stable if and only if
Impulse response is absolute summable

 hk   
k = −
Let’s write the output of the system as
yn =
If the input is bounded


k = −
k = −
 hk xn − k    hk  xn − k 
x[n]  Bx
Then the output is bounded by
yn  Bx

 hk 
k = −
The output is bounded if the absolute sum is finite
An LTI system is causal if and only if
hk  = 0 for k  0
Causality & Stability- Example
ℎ 𝑛 = 𝑎𝑛 𝑢 𝑛
Convolution Summation
yn =

 xk hn − k 
k = −
Note that the above operation is convolution, and can be written in short by y[n] =
x[n]  h[n].
The output of an LTI system is equivalent to the convolution of the input and the
impulse response.
In a LTI system, the input sample at n = k, represented as x[k][n-k], is transformed
by the system into an output sequence x[k]h[n-k] for − < n < .
Applications
Visualizing Convolution
There are four basic steps to the calculation:
Tabular Mechanism
∞
Convolution in the time domain:
𝑦 𝑛 = ෍ 𝑥[𝑘] ℎ[𝑛 − 𝑘]
𝑘=−∞
y[n] = 2 –3 3 3 –6 0 1 0 0
Example: Tabular
𝑥 𝑛 = 1, 2, 3
ℎ 𝑛 = {1, 2, 3}
Example: Convolution Summation
𝑥 𝑛 = 1, 2, 3
ℎ 𝑛 = {1, 2, 3}
Example:
The operation has a simple graphical interpretation:
Calculating Successive Values
• We can calculate each output point by shifting the unit pulse
response one sample at a time:

y[n] =
 x[k ] h[n − k ]
k = −
• y[n] = 0 for n < ???
y[-1] =
y[0] =
y[1] =
…
y[n] = 0 for n > ???
• Can we generalize this result?
Graphical Convolution (Cont.)
Observations:
y[n] = 0 for n > 4
If we define the duration of h[n] as the difference in time from the first nonzero
sample to the last nonzero sample, the duration of h[n], Lh, is
4 samples.
Similarly, Lx = 3.
The duration of y[n] is: Ly = Lx + Lh – 1. This is a good sanity check.
Graphically
x ( )
t
h (t )
h (t −  )
x ( )
t
Overlap
Example
DTC-FLIP & SLIDE method(Finite Length Signals)
Computation of Discrete Convolution
Convolution
Useful Summation
Convolution