Elementary signals

Elementary signals such as exponential and sinusoidal signals, the step function, the impulse function, and the ramp function feature prominently in the study of signals and systems. 

Here are various elementary signals discussed one by one.

Exponential signals

The general form to represent an exponential signal is:

x(t)=Be^at,

Here, both B and a are real parameters. B represents the amplitude of the exponential signal measured at time t=0.

Depending on the other parameter a is positive or negative, we may identify two special cases:

• Decaying exponential, where a<0
• Growing exponential, where a>0.

The exponential nature of a signal is confirmed by:

r=e^a, for some a. 

Sinusoidal signals

The general form to represent sinusoidal signals is:

x(t)=Acos(wt+o),

Where A is the amplitude, w is the frequency in radians per second and o is the phase angle in radians. A sinusoidal signal is an example of a periodic signal, the period of which is:

T=2pi/w

We may readily show that:

x(t+T)=Acos(w(t+T)+o)
=Acos(wt + wT + o)
=Acos(wt + 2 pi+ o)
=Acos(wt+o)
=x(t). 

Which satisfies the condition for a periodic signal. 

The discrete-time signal may or may not be periodic. For it to be periodic with N samples it must yield:

x[n+N] = Acos((omega)n + (omega)N + o)

Which will require,

omegaN=2pim radians,
omega=2pim/N radians/cycle, integer m, N. 

Relation between sinusoidal and complex exponential signals

Let us consider a complex exponential

e^iθ.

Using Euler’s identity, we can expand this term as:

e^iθ=cosθ+jsinθ

According to the result, we can express the continuous-time sinusoidal signal of the above equation is the real part of the complex exponential signal Be^iwt, 

Where B=Ae^iΦ. This quantity in itself is complex, therefore it can be further written as:

Acos(wt+Φ)=Re{Be^iwt}. 

The Re{} denotes the real part of the complex quantity enclosed inside the braces. 

We can also define a continuous-time sinusoidal in terms of the sine function, such as:

x(t)=Asin(wt+Φ).

It is represented by the imaginary part of the complex exponential signal 

Be^iwt.

We can write it as:

Asin(wt+Φ)=Im{Be^iwt}.

Where Im denotes the imaginary part of the complex quantity enclosed inside the braces. The sinusoidal signal of x(t)=Asin(wt+Φ) differs from x(t)=Acos(wt+o), by a phase angle of 90 degree. 

Similarly, in a discrete-time case, we can write:

Acos(Ωn+Φ)=Re{Be^iΩn}
And, A sin(Ωn+Φ)=Im{Be^iΩn}. 

Where B is defined in terms of A and Φ.  

Exponentially damped sinusoidal signals

The multiplication of a sinusoidal signal by a real-valued decaying exponential signal results in a new signal known as an exponentially damped sinusoidal signal. Multiplying the continuous-time sinusoidal signal Asin(wt+Φ by the exponential e^-αt results in the exponentially damped sinusoidal signal

x(t)=Ae^-αt sin(wt+Φ), α>0.

The discrete-time version for an exponentially damped sinusoidal signal is given by:

x[n]=Br^nsin[Ωn+Φ]. 

The parameter range will be 0<|r|<1. 

Step function

The unit step function for the discrete time version is defined by:

u[n]=1, n>=0 or u[n]=0, n<0.

The continuous time version of the unit-step signal is:

u[t]=1, t>0 or u[t]=0, t<0. 

In electronics, it is useful as the output of a system due to step input can tell us a lot about how quickly a system will respond to an abrupt change in the input. The unit step function u(t) is a simple function to apply. In a battery or a dc source is applied at t=0. For example, closing a switch. 

It can also be used to construct other discontinuous waveforms, for example, a rectangular pulse. 

Impulse function

The discrete time version of unit impulse is given by:

δ[n]=1, n=0 or δ[n]=0, n!=0.

The continuous-time version of unit impulse is given by:

δ[t]=0, t !=0. 

Ramp function

As we have seen earlier the impulse function δ[t] is the derivative of the step function u(t) with respect to time. Similarly, the integral of the step function u(t) is a ramp function of a unit slope. It is defined as:

r(t)=1, t>=0 or r(t)=0, t<0.
Or, r(t)=tu(t).

The ramp function enables us to evaluate how a continuous-time system would respond to a signal that increases linearly with time. 

Graph for ramp function of a unit slope. 

Reference

Elementary signals