0.5 Lecture 6:continuous time fourier transform (ctft)

Lecture #6:

CONTINUOUS TIME FOURIER TRANSFORM (CTFT)

Motivation:

• Extends the notion of the frequency response of a system to the frequency content of a signal.
• Widely used tool in many areas (communications, control, signal processing, X-ray diffraction, Medical imaging — CAT&PET scan).

• Continue development of Fourier transform pairs.
• Illustrate different methods for finding Fourier transforms

Outline:

• The continuous time Fourier transform (CTFT)
• Properties of the CTFT
• Simple CTFT pairs

• Fourier transform pairs

– Fourier transform of the unit step function

– Fourier transform of causal sinusoids

– Fourier transform of rectangular pulse — the sinc function

– Fourier transform of triangular pulse

• Filtering the ECG revisited
• Conclusion

Example — How to filter the ECG?

The recorded activity from the surface of the chest includes the electrical activity of the heart plus extraneous signals or “noise.” How can we design a filter that will reduce the noise?

It is most effective to compute the frequency content of the recorded signal and to identify those components that are due to the electrical activity of the heart and those that are noise. Then the filter can be designed rationally. This is one of many motivations for understanding the Fourier transform.

I. THE CONTINUOUS TIME FOURIER TRANSFORM (CTFT)

1/ Definition

The continuous time Fourier transform of x(t) is defined as

$X\left(f\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{j2\pi ft}}dt$

and the inverse transform is defined as

$x\left(t\right)={\int }_{-\infty }^{\infty }X\left(f\right)e^{\mathrm{j2\pi ft}}df$

2/ Relation of Fourier and Laplace Transforms

The bilateral Laplace transform is defined by the analysis formula

$X\left(s\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{st}}dt$

and the inverse transform is defined by the synthesis formula

$x\left(t\right)=\frac{1}{\mathrm{j2\pi }}{\int }_{C}X\left(s\right)e^{\mathrm{st}}ds$

Now if the jω axis is in the region of convergence of X(s), then we can substitute s = jω = j2πf into both relations to obtain

$X\left(\mathrm{j2\pi f}\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{j2\pi ft}}dt$

$x\left(t\right)=\frac{1}{\mathrm{j2\pi }}{\int }_{-\mathrm{j\infty }}^{\mathrm{j\infty }}X\left(\mathrm{j2\pi f}\right)e^{\mathrm{j2\pi ft}}d\left(\mathrm{j2\pi f}\right)$

3/ Form of the Fourier transform

Finally, by canceling j2π and changing the variable of integration from j2πf to f we obtain

$X\left(\mathrm{j2\pi f}\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{j2\pi ft}}dt$

$x\left(t\right)={\int }_{-\infty }^{\infty }X\left(\mathrm{j2\pi f}\right)e^{\mathrm{j2\pi ft}}df$

It is clumsy to write X(j2πf). Therefore, from this now on, we rewrite the function X(j2πf) described above as in a simpler form X(f) such that the transform pair would be symmetrical.

X(j2πf) Þ X(f)

The new Fourier transform pair will be in the form

$X\left(f\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{j2\pi ft}}dt$ $x\left(t\right)={\int }_{-\infty }^{\infty }X\left(f\right)e^{\mathrm{j2\pi ft}}df$

4/ Notation

Notation for the Fourier transform varies appreciably from text to text and in different disciplines. Another common notation is to define the Fourier transform in terms of jω as follows

$X\left(\mathrm{j\omega }\right)={\int }_{-\infty }^{\infty }x\left(t\right)e^{-\mathrm{j\omega t}}dt$ $x\left(t\right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(\mathrm{j\omega }\right)e^{\mathrm{j\omega t}}d\omega$

Differences in notation are largely a matter of taste and different notations result in different locations of factors of 2π. The notation we use minimizes the number of factors of 2π that appear in the expressions we will use in this subject and makes the duality of the Fourier transform with its inverse more transparent.