1. Lecture 3 Discrete time systems

2. Representation of discrete- time systems
x [n]
T{.}
y [n]
x[n]
y[n]
Example: Ideal delay system
y [n] = x [n – nd] -∞ < n < ∞

3. The accumulator system
A memoryless system
The accumulator system

4. Moving average
It is a low pass filter!!!

5. Memory Causality Stability Time invariance Linearity
Systems properties
Memory
Causality
Stability
Time invariance
Linearity

6. Memory: Causality A system is memoryless if y[n] = f ( x[n] )
i.e. it sees only present values.
A system has memory if y [n] depends on previous values
it can also depend on present and future values!
Causality
A system is causal if the output y[n] depends only on present and/or past values.
On-line systems are causal by definition

7. Time-invariance Stability
A system is time invariant if a shift in the input causes a corresponding shift of the output.
For all n0 x1 [n] = x [n-n0] gives y1[n] = y [n-n0]
Stability
A system is stable if every bounded input sequence produces a bounded output
i.e. it never diverges.
If |x[n]| ≤ Bx < ∞ then |y[n]| ≤ By < ∞

8. Linearity 1) Additive property. 2) Scaling property
Linear systems obey the principle of superposition.
1) Additive property.
T {x1[n] + x2[n]} = T {x1[n]} + T {x2[n]} = y1[n] + y2[n]
2) Scaling property
T {a x1[n]} = a T {x1[n]} = a y[n]
Altogether: T {a x1[n] + b x2[n]} = a T{x1[n]} + b T{x2[n]}
More generally:
If x[n] = Σk ak xk[n] then y[n] = Σk ak yk[n]
where yk[n] is the system response to the input xk[n]

9. Exercise:
Which properties (linearity, causality, time-invariance, stability and memory) posses the following systems:
a) y[n] = 3 x[n] – 4 x[n-1]
b) y[n] = 2 y[n-1] + x[n+2]
c) y[n] = n x[n]
d) y[n] = cos (x[n])
e) y[n] = log10 (x[n])
f) y[n] = x[n]4
g) The accumulator system
h) The ideal delay system
i) The moving average system

10. A sequence can be represented as a linear combination of delayed impulses:

11. Linear time-invariant systems (LTI)
Let hk[n] be the response to d[n-k] (an impulse at n = k)
If the system is linear
If the system is time-invariant

12. Convolution sum
Convolution sum
Linear time-invariant systems can be described by the convolution sum!

13. Note that:
A linear time-invariant system can be completely characterized by its input response h[n]
d[n]
h[n]
LTI

14. Properties of LTI systems (or properties of convolution)
Convolution is conmutative
x[n]  h[n] = h[n]  x[n]
Convolution is distributive
x[n]  (h1[n] + h2[n]) = x[n]  h1[n] + x[n]  h2[n]

15. Cascade connection:
y[n] = h1[n]  [ h2[n]  x[n] ] = [ h1[n]  h2[n] ]  x[n]
x[n] h2 h1
y[n] =
x[n]
h1h2
y[n]

16. Parallel connection
y[n] = h1[n]  x[n] + h2[n]  x[n] ] = [ h1[n] + h2[n] ]  x[n] h1
x[n] +
y[n] h2 =
x[n]
h1+h2
y[n]

17. LTI systems are stable iff
LTI systems are causal if
h[n] = n < 0