Classification of Systems - Digital Signal Processing, Engineering - Computer Science Engineering (CSE) PDF Download

Classification of Systems

1. a. Static systems or memory less system. (Non Linear / Stable)

Ex. y(n) = a x (n)

= n x(n) + b x³(n)

= [x(n)]² = a(n-1) x(n)

If its o/p at every value of ‘n’ depends only on the input x(n) at the same value of ‘n’ Do not include delay elements. Similarly to combinational circuits.

b. Dynamic systems or memory.

If its o/p at every value of ‘n’ depends on the o/p till (n-1) and i/p at the same value of ‘n’ or previous value of ‘n’.

Ex. y(n) = x(n) + 3 x(n-1)

= 2 x(n) - 10 x(n-2) + 15 y(n-1)

Similar to sequential circuit.

2. Ideal delay system. (Stable, linear, memory less if nd=0)

Ex. y (n) = x(n-nd)

nd is fixed = +ve integer.

3. Moving average system. (LTIV ,Stable)

y(n) = 1/ (m1+m2+1)   

This system computes the n^th sample of the o/p sequence as the average of (m₁+m₂+₁) samples of input sequence around the nth sample.

If   M1=0; M2=5

= 1/6 [x(7) + x(6) + x(5) + x(4) + x(3) + x(2)]

y(8) = 1/6 [x(8) + x(7) + x(6) + x(5) + x(4) + x(3)]

So to compute y (8), both dotted lines would move one sample to right.

4. Accumulator.      ( Linear , Unstable )

y(n) =

= y(n-1) + x(n)

x(n) = { …0,3,2,1,0,1,2,3,0,….}

y(n) = { …0,3,5,6,6,7,9,12,12…}

O/p at the n^th sample depends on the i/p’s till nth sample 

Ex: 

x(n) = n u(n) ; given y(-1)=0.  i.e. initially relaxed.

y(n) = 

5. Linear Systems.

If y₁(n) & y₂(n) are the responses of a system when x₁(n) & x₂(n) are the respective inputs, then the system is linear if and only if 

=  y1(n) + y2(n)   (Additive property)

  (Scaling or Homogeneity)

The two properties can be combined into principle of superposition stated as

Otherwise non linear system.

6. Time invariant system.

Is one for which a time shift or delay of input sequence causes a corresponding shift in the o/p sequence.

TIV
≠                                 TV

7. Causality.

A system is causal if for every choice of n_o the o/p sequence value at index n= n_o depends only on the input sequence values for n ≤no.

y(n) = x(n) + x(n-1) causal.

y(n) = x(n) + x(n+2) + x(n-4)  non causal.

8. Stability.

For every bounded input |x(n)|≤B_x <∞ for all n, there exists a fixed +ve finite value By such that |y(n)|≤ B_y <∞