Download + by(n-1)

Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Intelligent Systems Laboratory
Digital Signal Processing
Prof. George Papadourakis, Ph.D.
Discrete Signals and Systems
Sequences
• Discrete time signals are number sequences
 x is a sequence
 x(n) is a sample of the sequence at time n
Discrete Signals and Systems
Sequences
• Creation of a sequence
1.
Number creation and place them in a sequence
1,2,3,…,{N-1}
x(n) = n
1<n<N
2.
Iterative process
x(n) = x(n-1)/2,
x(0) = 1
3. Sampling of analog signal. Signal value create a sequence
A/D converter
1,2 time independent, 3 time dependent
Discrete Signals and Systems
Sequences
• Discrete function delta δ(n)
It has one non-zero element
• Very important input sequence in a digital system. The output sequence is
called impulse response and contains important information concearning the
behavior of the system.
• Discrete function delta delayed δ(n-k)
Discrete Signals and Systems
Sequences
• Unit step u(n)
• u(n) is related to δ(n)
u(n) – u(n-1) = δ(n)
Discrete Signals and Systems
Sequences
• Exponential Sequences :
x(n) = Aasn = Aa(σ + jw0)n
• s is a complex number
s = (σ + jw0)
• If s is a real number and a=e then the sequence is called real exponential
x(n) = Ae-|σ|n
Discrete Signals and Systems
Sequences
• Geometric Sequence :
real exponential sequence defined as :
Discrete Signals and Systems
Sequences
• Sinusoidal Sequence :
Discrete Signals and Systems
Properties of Sequences
• Sum of two signals :
w=x+y
w(k) = x(k) + y(k)
• Multiplication of two signals :
w = xy
w(k) = x(k)y(k)
• Multiplication of signal by a scalar :
w = cx
w(k) = cx(k)
• Energy of the signal :
• If a signal is delayed by m time units then x(k) becomes x(k-m)
• Sequence : Sum of scaled, delayed unit samples :
Discrete Signals and Systems
Properties of Sequences
Example :
Discrete Signals and Systems
Signal Measures
• The signal norm is defined :
• Some properties of the signal norm are:
Discrete Signals and Systems
Linear
• Norm 1 : Sum of the magnitudes of each signal sample
Used to determine system stability.
• Norm 2 : Provides a measure of the signal power.
It is the most frequent used measure.
• Norm infinity : Gives the peak magnitude of the signal.
Discrete Signals and Systems
Linear Shift-Invariant Systems
Discrete-time system : Converting input sequence x = x(n) into output sequence
y=y(n) through transformation φ[.]
A linear system is defined by the principle of superposition.
If,
Then a system is linear if and only if,
Discrete Signals and Systems
Linear Shift-Invariant Systems
Example 1 :
Is the following system linear?
y(n) = 10x(n) – 5y(n-1)
Solution :
αφ[x1(n)] = α10x1(n) – α5y1(n-1)
bφ[x2(n)] = b10x2(n) – b5y2(n-1)
αφ[x1(n)] + bφ[x2(n)] = α10x1(n) – α5y1(n-1) + b10x2(n) – b5y2(n-1)
φ[αx1(n) + b x2(n)] = 10[αx1(n) + bx2(n)] – 5[ αy1(n-1) + by2(n-1)]
Yes, the system is linear!
Discrete Signals and Systems
Linear Shift-Invariant Systems
Example 2 :
Is the following system linear?
y(n) = [x(n)]2
Solution :
αφ[x1(n)] = α[x1(n) ] 2
bφ[x2(n)] = b[x2(n) ] 2
αφ[x1(n)] + bφ[x2(n)] = α[x1(n) ] 2 + b[x2(n) ] 2
φ[αx1(n) + b x2(n)] =
= [αx1(n) + bx2(n)] 2 = α2[x1(n)] 2 + b 2[x2(n) ] 2 +2αbx1(n)x2(n)
No, the system is not linear!
Discrete Signals and Systems
Linear Shift-Invariant Systems
A system is time-invariant or shift-invariant if,
y(n) is response to x(n) then
y(n-k) is response to x(n - k)
,z-k : a signal delay of k samples
Example 1 :
Is the following system shift-invariant?
y(n) = 10x(n) - 5y(n-1)
Solution:
Yes, the system is shift-invariant!
Discrete Signals and Systems
Linear Shift-Invariant Systems
Example 2 :
Is the following system shift-invariant?
y(n) = nx(n)
Solution :
No, the system is not shift-invariant!
We said that we can express :
The system output response is :
Discrete Signals and Systems
Linear Shift-Invariant Systems
If the system is linear, the response of the system to a sum of inputs is the same
as the sum of the system’s responses to each of the individual inputs :
By definition :
If the system is shift-invariant :
φ[δ(k)] = h(k)
φ[δ(n-k)] = h(n-k)
If a system is linear and shift-invariant,
the convolution sum applies y(n) = x(n) * h(n)
Discrete Signals and Systems
Linear Convolution
The graph method of computing the Convolution Sum
• Folding one of the sequences x(n) or h(n) over the horizontal axis and getting
x(-k) or h(-k)
• Shifting the folded sequence creating x(n-k) or h(n-k)
• The addition of the product of the two sequences at time n yields the output
y(n)
Example :
What is the response y(n)
if h(n) = {1,2,3}
and x(n) = {3,1,2,1}
Discrete Signals and Systems
Linear Convolution
As a result, y(n) = {3,7,13,8,8,3}
• If x(n) : N samples, h(n) : M samples,
y(n) : N + M – 1 samples
Discrete Signals and Systems
Stability and Causality
• A system is stable if a bounded input produces a bounded output.
• Necessary and sufficient condition
• This is the norm as defined in a previous session
Example :
Is the following system stable :
y(n) = x(n) + by(n-1)
Calculating the norm we have :
The system is stable!
Discrete Signals and Systems
Stability and Causality
• A causal system is a system that at the time m produces a system output that
is dependent only on current and past inputs, that is n<m
• This is always true for a unit impulse response, it is zero for n<0
• A discrete-time, linear, shift-invariant system is causal if and only if
h(n) = 0 for n<0
Example :
Is the following system causal:
y(n) = x(n) + by(n-1)
Since the unit-sample response is zero for n<0..
The system is causal!
Discrete Signals and Systems
Digital Filters
• A broad class of digital filters are described by linear, constant coefficient and
difference equations
{ai} {bi} characterize the system ,
• Given initial conditions x(i), y(i), I = -1,-2,…,-M
input sequence x(n)
output sequence y(n) ,
• The system is causal.
• The system is Mth-order.
• Two main classes of digital filters :
 Infinite Impulse Response (IIR)
 Finite Impulse Response (FIR)
Discrete Signals and Systems
Digital Filters
• Infinite Impulse Response (IIR) : Current and past input samples and
past output samples
Example :
Determine impulse response for the first-order IIR filter y(n) = x(n) + by(n-1)
Assume : x(n) = 0, y(n) = 0 for n<0
x(n) = δ(n)
h(n) = δ(n) + bh(n-1)
h(n)=0
n<0
h(0)=1 + b 0 = 1
h(2)=0 + b b = b2
……..
h(1)=0 + b 1 = b
h(3)=0 + b b2 = b3
h(n)= bn u(n)
Discrete Signals and Systems
Digital Filters
• Finite Impulse Response (FIR) : Current and past input samples.
• The coefficients of the FIR filter are equivalent to the filter’s impulse
response.
Why?
Remember convolution?
h(k) = bk
h(k) = 0
k = 0,1,2,…,M
otherwise
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Intelligent Systems Laboratory