The convolution integral and its evaluation procedure
The convolution integral
The output of a continuous-time LTI system can also be determined using the input and the system’s impulse response. This approach and result are analogous to those in the discrete-time case.
Firstly, we will express a continuous-time signal as the weighted superposition of time-shifted impulse:
x(t) = ∫ x(T)δ(t-T) dT.
In the above equation, the superposition is an integral instead of a sum and the time shifts are given by the continuous variable T. The weights of x(T)dT are derived from the value of the signal x(t) at the time T at which each impulse occurs.
Let us take an operator H, that denotes the system to which the input x(t) is applied.
We consider the system output in response to a general input expressed as the weighted superposition:
y(t)=H{x(t)} =H{∫ x(T)δ(t-T) dT}.
Applying the linearity property of the system. We can interchange the order of the operator H and integration to obtain the following:
y(t) = ∫ x(T)H{δ(t-T) dT}.
In the discrete-time case, the response of a continuous-time linear system to time-shifted impulses completely describes the input-output characteristics of the system.
Then, we can define the impulse response h(t) = H{δ(t-T ) = h(t-T).
That is, the time invariance implies that a time-shifted impulse input generates a time-shifted impulse response output.
It can be also seen in the following diagrams:
The image is taken from swarthmore.edu .
Convolution integral evaluation procedure
Similar to the evaluation of convolution sum, the procedure for evaluating the convolution integral is based on defining an intermediate signal that simplifies the evaluation of the integral.
The convolution integral is expressed as:
x(t) = ∫ x(T)δ(t-T) dT.
It can be redefined as the intermediate signal:
wt(T) = x(T)h(t-T) dT.
In this equation, the T is the independent variable and the time t is treated as a constant. This is indicated by writing t as a subscript and T within the parentheses of wt(T).
Therefore, h(t-T) = h(-(T-t), is reflected and shifted (by -t) version of h(T).
If t<0, then the h(-T) is time-shifted to the left.
And if t>0, then h(-T) is time-shifted to the right.
The time shift t determines the time at which the output of the system is evaluated. Therefore, the equation becomes:
y(t) = ∫ wt(T) d T
Therefore, the system output at any time t is the area under the signal wt(T).
Generally, the mathematical representation of wt(T) depends on the value of t.
In the discrete-time case, we avoid evaluating the equation: y(t) = ∫ wt(T) d T, at an infinite number of values of t by identifying the intervals of t at which wt(T) does not change.
Therefore, we will only need to evaluate y(t) = ∫ wt(T) d T using wt(T) associated with each interval. It is usually beneficial to graph both x(T) and h(t-T) in determining wt(T) and identifying the appropriate set of shifts.
The entire procedure can be summarized as follows:
Step 1: Graph the x(T) and h(t-T) as a function of the independent variable T. To obtain h(t-T), reflect h(T) about T=0 to obtain h(-T), and then shift h(-T), by -t.
Step 2: Begin with the shift t large and negative, that is, shift h(-T) to the far left on the time axis.
Step 3: Write the mathematical expression for wt(T).
Step 4: Increase the shift t by moving h(t-T) towards the right until the mathematical representation of wt(T) changes. The value of t at which the change occurs defines the end of the current set of shifts and the beginning of a new set.
Step 5: Let t be in the new set. Repeat steps 3 and 4 until all the sets of the shifts t and the corresponding representations of wt(T) are identified. This usually implies increasing t to a large positive value.
Step 6: For each set of shifts t, integrate wt(T) from T = -infinity and and T = + infinity to obtain y(t).
The effect of increasing t from a large negative value to a large positive value is to slide h(-T) past x(T) from left to right.
Transitions in the set of t are associated with the same form of wt(T). It generally occurs when a transition in h(-T) slides through a transition in x(T).
We can easily identify these intervals by graphing h(t-T) beneath x(T).
Reference