Test: Discrete Time Signals

1) Functional Representation.

Example:

2) Tabular method of representation

Example:

3) Sequence Representation

Example:

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Recursive discrete time system:

If a system output y(n) at time n depends on any number of past output value y(n – 1), y(n – 2), …, it is called a recursive system.

This system requires two multiplication, one addition, and one memory location. This is a recursive system which means the output at time n depends on any number of a past output values.

So, a recursive system has feedback output of the system into the input. This feed back loop contains a delay element.

Non-Recursive discrete time system:

If y(n) depends only on the present and past input, it is called non-recursive. It does not exhibit the necessity of any feedback.

Concept:

Output = Input × Eigen value.

Calculation:

Let, Input x(n) =  (-0.5)ⁿ

h(n) = 2ⁿ u(2n + 2)

= 2ⁿ u(n + 1)

= 2ⁿ⁺¹ u(n+1) / 2

(scaling does not affect the discrete-time signal)

putting -0.5 in place of Z, we get:

= -1/20

-1/20 is the eigenvalue

Output
 

Concept:

The signal u[n] is defined as:

u[n] = 1 for n ≥ 0

u[n] = 0 for n < 0

Also, the shifted version of u[n] is defined as:

u[n - n₀] = 1 for n ≥ n₀

u[n - n₀] = 0 for n < n₀

Now the combination of two discrete time signals can be analysed as:

u[n] - u[n - n₀] = 1 for 0 ≤ n ≤  n₀ - 1

u[n] - u[n - n0] = 0, otherwise

Application:

Since the given signal is defined as:

x[n] = 1, for 0 ≤ n ≤ 5

We can write:

x[n] = u[n] - u[n - 6]

Concept:

Linearity: Necessary and sufficient condition to prove the linearity of the system is that the linear system follows the laws of superposition i.e. the response of the system is the sum of the responses obtained from each input considered separately.

y{ax₁[t] + bx₂[t]} = a y{x₁[t]} + b y{x₂[t]}

Conditions to check whether the system is linear or not.

• The output should be zero for zero input
• There should not be any non-linear operator present in the system.

Causal system:

If O/P of the system is independent of the future value of input then the system is said to be causal. Causal systems are practical or physically reliable systems.

Analysis:

y(n) = [x(n)]2 

Linearity check:

x₁(n) → x₁(n)²

x₂(n) → x₂(n)²

x₃(n) = [x1(n) + x₂(n)] →  [x1(n) + x2(n)]² = x₁(n)² + x₂(n)² + 2x₁(n) x₂(n)

≠ x1(n)² + x₂(n)²

∴ Non - linear

Causality check:

y(0) = x(0)²

y(1) = x(1)²

y(-1) = x(-1)²

∴ the system is causal.

The arrow indicates the origin. This can be represented as:

In terms of unit impulse sequences, this can be represented as:

x(n) = 2δ(n + 1) + 4δ(n) + 3δ(n – 2)

If the above recursive definition is repeated for all n, starting from 1,2.. then y[n] will be the sum of all x[n] ranging from 1 to n, making it an accumulator system.

As the value of the function depends solely on the value of the input at a time presently and/or in the past, it is a causal system.

The system would be unstable, as the output will grow out of bound at the maximally worst possible case.

A discrete-time signal is continuous in amplitude and discrete in time. It can either be present in nature or is sampled from an analog signal. A digital signal is discrete in amplitude and time.

Product of discrete-time signals is computed element by element.
⇒ x(n) = x₁ (n) * x₂ (n) = {2×1, 1×1.5, 1.5×0, 3×2} = {2,1.5,0,6}.

It is BIBO stable in nature, i.e. bounded input-bounded output stable.

Unit step is defined by: x(n) = 1 for n ≥ 0 and x(n) = 0 for n < 0.

∑^∞_{n= -∞} δ(n + 3)(n² + n)
⇒ δ(n + 3)=1 when n= -3 otherwise 0.
Therefore, the limit reduces to n = -3
⇒ ∑^∞_{n = -∞} δ(n + 3)(n² + n) = (n² + n)|_{n = -3} = (-3)²-3 = 9 – 3 = 6.

Unit impulse function can be obtained by using a limiting process on the rectangular pulse function. Area under the rectangular pulse is equal to unity.