An Overview of The Differences between Time Domain and Frequency Domain

In Control System Engineering, the response of a system is analyzed for designing a suitable controller. The controller is needed to control the system behaviour as per requirements. As technology is moving toward automation, the role of sensors and actuators play a crucial role in any engineering field. 

Sensors sense the system output and send the signal to the controller. The controller gives the command to the actuator so that the manipulated input to the system will be varied. In order to design a suitable controller for a system, the characteristics of the system need to be analyzed first. Hence, the system response will be studied either in the time domain or frequency domain.

Time Domain

The time domain graph depicts how the signal varies with respect to time. In this time-domain graph, the magnitude of the signal is represented at each instant time. If it is a continuous system, the signal will be represented as continuous. If it is a discrete system, the magnitude of the signal will be represented in distinct time intervals. The Cathode Ray Oscilloscope(CRO)  device is the common device used to analyse the signal in a time domain.

Frequency Domain

The frequency domain graph shows how much of a signal lies within each frequency domain. Here in the frequency domain, the signal is represented by the sum of sinusoidal waves of different frequencies and each with a certain amplitude. Frequency domain analysis is very common in Control system Engineering. Thus, this refers to analyzing the signals with respect to the frequency of each signal.

1. Signals are represented in the time domain
2. Signals are represented in the frequency domain
3. Relationship between time and frequency domain.

In the time domain, the amplitude of the signal is plotted against the time and the frequency of the signal is untold. In the frequency domain, the amplitude of the signal is plotted against the frequency whereas how the signal varies with time is not given. Hence, each domain of signal representation provides us with different kinds of information about the same signal. Based on the objective, the analysis in one domain is advantageous over the other. The following methods are used to transform from the time domain to the frequency domain.

• Aperiodic Signal: The signal that does not repeat itself after a specific interval of time. Fourier transform is employed to transform such signals from the Time domain to the Frequency domain.
• Periodic Signal: Periodic signal is nothing but has the same pattern of occurrence over a definite time frame. Fourier series is used if the signal is periodic
• Continuous and Aperiodic signal: The signal which is having some amplitude at any instant of time but the recurrence of the pattern will not be seen. 
• Discrete Signal: If the signal has amplitude only at a specific time interval, then it is called Discrete Signal. Laplace transform and Z-transform are used to transform Continuous and Discrete Signals respectively.

Fourier Transformation Method:

Signals, given in one domain, can be transformed to another domain using a mathematical transformation method called the Fourier formula (Fourier transform and Fourier series). Fourier stated that any signal in the time domain can be represented as a summation of sinusoidal waveforms of different frequencies. Sinusoidal waveforms are used because they do not change their shape when they pass through an LTI (Linear Time Invariant) system. x(t) is the signal in the time domain that can be transformed to the frequency domain by Fourier transform. It is given by,

Laplace Transformation Method:

Laplace transform decomposes the signals in the time domain into a domain of both sine and exponential functions. It is otherwise called an s-domain. The sine wave frequency wave is described by ω, and the amplitude is described by 𝛔. Therefore,

x(t) is the signal in the time domain that can be transformed to the frequency domain by Laplace transform and it is given by,

Unlike Fourier transform, Laplace transform gives information about the magnitude gain of a signal. Hence, the Laplace transform of the signal is used in control system analysis. The application of the Laplace transform makes a differential equation into an algebraic equation that is much easier to manipulate and to design a suitable compensator for systems. z-Transform is used for discrete systems. It is considered a discrete-time equivalent of the Laplace transform.