Introduction to time-domain representation, convolution sum, and evaluation procedure.  

The convolution sum

To begin explaining the convolution sum we start by assuming a discrete-time case. 

At first, an arbitrary signal is expressed as a weighted superposition of shifted impulses. Then, the convolution sum is found by applying a signal represented in an LTI system. A similar procedure is used to find convolution integral later on. 

Let the signal x[n] be multiplied by the impulse sequence δ[n].

Implying: x[n]δ[n] = x[0]δ[n].

The relationship is generalized to the product of x[n] and a time-shifted impulse sequence to obtain:

x[n]δ[n-k] = x[k]δ[n-k].

Where n represents the time index; therefore x[n] denotes the entire signal, while x[k] represents a specific value of x[n] at time k. 

The multiplication of a signal by a time-shifted impulse results in a time-shifted impulse with amplitude given by the value of the signal at the time the impulse occurs. 

It allows us to express x[n] as the following weighted sum of time-shifted impulses:

x[n]=........+x[-2]δ[n+2] +x[-1]δ[n+1]+ x[0]δ[n]......

This equation can be rewritten as:

x[n]=Σ x[k]δ[n-k].

The graphical illustration of convolution sum is given below:

Image is taken from electrical academia. 

Let us now use the H operator to write the equation:

y[n]=H{x[n]}
      =H{Σ x[k]δ[n-k]}. 

Now using the linearity property to interchange the system operator H with the summation, we obtain:

y[n]=Σ H{x[k]δ[n-k]}. 

As n is the time index, the quantity x[k] is a constant with respect to the system operator H. Using linearity again, we can interchange H with x[k] to obtain:

y[n]=Σ x[k]H{δ[n-k]}. 

It indicates that the system output is a weighted sum of the response of the system to time-shifted impulses. This response completely characterizes the system’s input-output behavior and is a fundamental property of the linear system. 

Moving ahead, if we assume that the system is time-invariant, then a time shift in the input results in a time shift in the output. 

It implies that the output due to a time-shifted impulse is a time-shifted version of the output. 

The impulse is given by:

H{δ[n-k]}=h[n-k]

Therefore, the output of an LTI system is given by a weighted sum of time-shifted impulse responses.

It is a direct consequence of expressing the input as a weighted sum of time-shifted basis functions. 

The convolution sum equation is given below:

x[n]*h[n]=Σ x[k]h[n-k].

The input will be decomposed as a sum of weighted and time-shifted unit impulses, with the kth impulse input represented in the right half as:

H{x[k]δ[n-k]}=x[k]h[n-k]. 

The output component is obtained by shifting the impulse response k units in time and multiplying by x[k]. The total output y[n] in response to the input x[n] is obtained by finding the summation of all the individual outputs:

y[n]=Σ x[k]h[n-k]. 

For each value of n, we find the sum of the outputs associated with each weighted and time-shifted impulse input from k=-infinity to +inifnity. 

Convolution sum evaluation procedure

As mentioned earlier convolution sum is expressed as:

y[n]=Σ x[k]h[n-k]. 

Let us suppose, we define the intermediate signal as:

wn[k]=x[k]h[n-k]. 

as the product of x[k] and h[n-k]. In this definition, k is the independent variable and we explicitly indicate that n is treated as a constant by writing n as a subscript. 

Now as we know, 

h[n-k] = h[-(k-n)], it is reflected and time shifted version of h[k]. 

Therefore, if n is negative, then h[n-k] is obtained by time shifting h[-k] to the left. 

While if the n is positive then h[-k] is shifted to the right. This time shift determines the time at which we evaluate the output of the system, as:

y[n]=Σ wn[k]. 

Though, we only need to determine one signal wn[k], for each time n at which we desire to evaluate the output. 

Procedure 1:

Reflect and shift convolution sum evaluation

Step 1: Graph both x[k] and h[n-k] as a function of the independent variable k. To determineh[n-k], first reflect h[k] about k=0 to obtain h[-k], then shift by -n. 

Step 2: Start with n being large and negative. 

Step 3: Write the mathematical expression to represent the intermediate signal i.e wn[k]. 

Step 4: Increase shift n toward the right until the mathematical representation for wn[k] changes. 

Step 5: Let n, be the new interval. 

Step 6: At every interval of time shifts, sum all the values of the corresponding wn[k] to obtain y[n] on that interval. 

Reference 

Introduction to time-domain representation, convolution sum, and evaluation procedure.