Question 6.10: CTFT using the differentiation property Find the CTFT of x(t......
CTFT using the differentiation property
Find the CTFT of x(t) = rect((t +1)/2) − rect((t −1)/2) using the differentiation property of the CTFT and the table entry for the CTFT of the triangle function (Figure 6.30).
The function x(t) is the derivative of a triangle function centered at zero with a base halfwidth of 2 and an amplitude of 2
x(t) = \frac{d}{dt} (2 tri(t/2 )).
In the table of CTFT pairs we find tri(t)\xleftrightarrow{\mathcal{F}}sinc²(f). Using the scaling and linearity properties, 2tri(t /2)\xleftrightarrow{\mathcal{F}}4sinc² (2 f). Then, using the differentiation property, x(t)\xleftrightarrow{\mathcal{F}} j8πf sinc² (2 f). If we find the CTFT of x(t) by using the table entry for the CTFT of a rectangle rect(t)\xleftrightarrow{\mathcal{F}}sinc(f) and the time scaling and time shifting properties we get x(t)\xleftrightarrow{\mathcal{F}} j4sinc(2 f)sin(2πf), which, using the definition of the sinc function, can be shown to be equivalent.
x(t) \xleftrightarrow{\mathcal{F}} j8πf sinc²(2 f) = j8πf sinc(2 f ) \frac{sin(2πf)}{2πf} = j4sinc(2 f)sin(2πf)