Test: Discrete Time Signals & Useful Signals - Electrical Engineering (EE) MCQ

‘y’ will be periodic only if x attains the same value after some time, T. However, if x is a one-one discrete function, it may not be possible for some x[n].

Using Euler’s rule, exp(2pi*n) = 1 for all integer n. Thus, the answer can be derived.

A time invariant system’s output should be directly related to the time of the output. There should be no scaling, i.e. y(t) = f(x(t)).

Since the function needs to store what it was at a time 4 units and 1 unit before the present time, it needs memory.

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

As we take the sum of y[n], terms cancel out and deem z[n] to be BIBO stable.

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 unit impulse function has value 1 at n = 0 and zero everywhere else.

The unit step function u[n] = 1 for all n>=0, hence u[1] = 1.

The numerator evaluates to 1, and the denominator is t, hence the answer is 1/t.

The first integral is 1, and the overall integral evaluates to t.

The function assumes the same value after t+2pi/w, hence the period would be 2pi/w.

The sin(t)and cos(t) can be found using Euler’s rule.

The expression {sum from -inf to inf} exp(jwn)*d[n] is equal to 1, since d[n] is equal to 1 when n = 0 and 0 otherwise. The sum can then be simplified to the form of a geometric series, which has a sum of 2.

The sum will exist only for n = 0, for which the product will be 1.

Only one of the values can be one at a time, others will be forced to zero, due to the delta function.

 None of the derivatives are defined at t=0.

Euler rule: exp(jt) = cos(t) + jsin(t).

Stability of LTI system:

A linear time-invariant (LTI) system is said to be stable if:

1. The bounded input sequence always produces a bounded output sequence.
2. Its natural response approaches zero as time approaches infinity.
3. All the poles of the system lie on the left-hand side of the jω axis.
4. The impulse response of an LTI system is absolutely summable i.e. 
   

The response of a stable LTI system is:

Deterministic signal:

• A signal is said to be deterministic if there is no uncertainty with respect to its value at any instant of time.
• The signals which can be defined exactly by a mathematical formula are known as deterministic signals.
• Example- sin (t), cos (t), e^at 

Non-deterministic signal:

• The behavior of these signals is random i.e. not predictable w.r.t time.
• There is an uncertainty with respect to its value at any time.
• These signals can't be expressed mathematically.
• Example: Thermal Noise generated is a non-deterministic signal.

Periodic signal:

• A periodic signal is a signal that repeats its values at regular intervals.
• A periodic is expressed as x(t) = x(t + t_o)
• Example- sin (t), cos (t)

​Non-Periodic signal:

• A non-periodic signal is any signal that does not repeat itself after any period of time.
• Example- speech signal, e^at

Concept:

Basic linear operations are given below-

• Multiplication of input signal with a constant value: y(t) = Ax(t)
• Time-shifting the input signal y(t) = x(t−t₁)
• Scaling of the input signal y(t) = x(At)
• Combinations of scaling, shifting and multiplication , e.g. y(t) = Ax(b(t−t₁))
• Summations of terms that are linear, e.g. y(t) = Ax(t)+x(t−t₁)
• Convolution of two input signal, eg y(t) = x(t)*h(t)

Some non-linear operations are given below:

• Multiplication of the signal with itself, i.e. y(t) = x(t)⋅x(t)
• Applying any non-linear function to the signal, e.g. y(t) = sin(x(t))
• Adding constant terms, which are independent of the signal, i.e. y(t) = x(t)+A