A signal's response to an impulse at the time instant t = 5 is represented by the output of a convolution between the signal x(t) and Dirac(t-5) (an impulse delayed by 5 units of time).
When you convolve a signal x(t) with Dirac(t-5) (an impulse delayed by 5 units of time), the output of the convolution represents the response of the signal x(t) to the impulse at the time instant t = 5. In other words, it measures how the signal x(t) "reacts" to the impulse at the specific time delay of 5 units.
Mathematically, the convolution of two functions x(t) and h(t) is defined as follows:
(x * h)(t) = ∫[x(τ) * h(t - τ)] dτ
In this case, x(t) is the input signal, and h(t) is Dirac(t-5), which is an impulse delayed by 5 units. The convolution (x * h)(t) represents how x(t) is influenced by the impulse at t = 5.
Now, regarding the statement about convolution in the time domain and frequency domain:
Convolution in the time domain is a mathematical operation that combines two signals in the time domain. It is represented by the convolution integral as shown above. It is used to model the output of a linear system when an input signal is applied.
Convolution in the frequency domain is a technique used to simplify the convolution operation when dealing with certain types of signals or systems. It takes advantage of the property that convolution in the time domain corresponds to multiplication in the frequency domain. This is known as the convolution theorem.
Mathematically, the convolution theorem states that the Fourier Transform of the convolution of two functions is equal to the product of their individual Fourier Transforms. In symbols: