Convolution of a discrete time system with a delta function

The convolution operation is a fundamental mathematical operation in signal processing and systems theory. It is used to combine two signals or systems to obtain a new signal or system that represents their interaction.

In the case of a discrete time system, the convolution of the system with a delta function is of particular interest. The delta function, denoted as δ[n], is a mathematical function that is zero everywhere except at n = 0, where it is infinite. In discrete time, it is often represented as a sequence of zeros with a single non-zero value of 1 at n = 0.

When we convolve a discrete time system with a delta function, we are essentially calculating the response of the system to the delta function. This can be interpreted as the system's impulse response, which characterizes how the system behaves when subjected to an instantaneous input.

Impulse response
The impulse response of a system is a fundamental property that describes how the system reacts to an impulse input. It represents the output of the system when the input is an idealized impulse signal, such as the delta function.

Convolution operation
The convolution of two sequences, x[n] and h[n], is defined as the sum of their element-wise products, where one of the sequences is flipped in time and shifted for each multiplication. Mathematically, the convolution of x[n] and h[n] is denoted as y[n] = x[n] * h[n].

Convolution with a delta function
When we convolve a discrete time system with a delta function, the resulting output sequence represents the system's response to the impulse input. Since the delta function is non-zero only at n = 0, the convolution reduces to a simple multiplication of the system's impulse response with the value of the input at n = 0.

Interpretation
The result of convolving a discrete time system with a delta function is the system's impulse response. This implies that the convolution with a delta function captures the essential characteristics of the system's behavior, as it represents the system's output when subjected to an instantaneous input.

Therefore, the correct answer is option 'B' - the convolution of a discrete time system with a delta function gives the system itself. It provides valuable insights into the system's properties and is widely used in the analysis and design of systems in various fields, including electrical engineering.