# Approximation of the Continuous Time Fourier Transform...

date post

13-Jan-2020Category

## Documents

view

0download

0

Embed Size (px)

### Transcript of Approximation of the Continuous Time Fourier Transform...

Approximation of the Continuous Time Fourier Transform

Signals & Systems

Lab 8

Continuous Time Fourier Transform (CTFT)

๐ ๐ = ๐ฅ ๐ก ๐โ๐2๐๐๐ก๐๐ก

โ

โโ

๐ฅ ๐ก = ๐ ๐ ๐๐2๐๐๐ก๐๐

โ

โโ

โUncleโ Fourier

Riemann Definite Integral

๐ ๐ฅ ๐๐ฅ

๐

๐

โ lim ฮ๐ฅโ0 ๐ ๐ฅ๐

โ ฮ๐ฅ๐

๐

๐=1

โUncleโ Riemann

Riemann Definite Integral

๐ ๐ฅ is defined where x โ ๐, ๐

ฮ๐ฅ๐ = ฮ๐ฅ = ๐โ๐

๐ called the subinterval (assumed all equal in our case)

n is the number of subintervals which โfillโ ๐, ๐

๐ฅ๐ โ are called the sample points of ๐ for each subinterval

As MATLAB can realistically operate only on discrete data we would like to use this definition to find an approximation to the CTFT.

๐ ๐ฅ ๐๐ฅ

๐

๐

โ lim ฮ๐ฅโ0 ๐ ๐ฅ๐

โ ฮ๐ฅ๐

๐

๐=1

The limit ฮ๐ฅ โ 0 means that the total interval ๐, ๐ will be filled with a infinite number of subintervals each with width ฮ๐ฅ!

๐ ๐ = ๐ฅ ๐ก ๐โ๐2๐๐๐ก๐๐ก

โ

โโ

โ lim ฮ๐กโ0 ๐ฅ ๐ก๐

โ ๐โ๐2๐๐๐ก๐ โ ฮ๐ก

๐

๐=1

In light of the previous observation we would like to express the Fourier Transform integral as a sum,

But in this form the expression for the Fourier transform is still impractical because it requires an infinite number of terms.

Now ๐ฅ ๐ก is a signal in continuous time. But in MATLAB it will take on amplitude values only at discrete time points which depend on when the continuous-time signal was sampled. So suppose ๐ฅ ๐ก is being sampled every ฮ๐ก seconds resulting in a sampled function ๐ฅ ๐ = ๐ฅ ๐ฮ๐ก where our time index, ๐, start at 0 (causal) and ends at N, the total number of samples.

Time-Domain Housekeeping

Hence the Fourier transform of this ๐ฅ[๐] signal is

Time-Domain Housekeeping

๐ ๐ = lim ฮ๐กโ0 ๐ฅ ๐ฮ๐ก ๐โ๐2๐๐๐ฮ๐กฮ๐ก

๐โ1

๐=0

Real signals of interest have finite ฮ๐ก time windows over the length of the signal, ๐ฮ๐ก, so the limit can be dropped resulting in the approximate expression

Careful this simple result may be deceiving! This result says that for a particular frequency ๐ a sum must computed. If X ๐ is to be specified at every possible frequency then the sum must be evaluated at every possible frequency! That is an enormous amount of computation especially if N is large.

Time-Domain Housekeeping

๐ ๐ โ ๐ฅ ๐ฮ๐ก ๐โ๐2๐๐๐ฮ๐กฮ๐ก

๐โ1

๐=0

The infinite number of really small subintervals can be replaced with a finite number of small subintervals.

Illuminating example when ๐ ๐ฅ is a real & even Function

๐ ๐ฅ โฑ ๐น ๐ Source: Bracewell

Explanation

Some authors will say that the Continuous-Time Fourier Transform of a function ๐ฅ ๐ก is the Continuous-Time Fourier Series of a function ๐ฅ ๐ก in the limit as ๐0 โ โ. This is equivalent to saying the Fourier Series can be extended to aperiodic signals. You can also think of the Fourier Transform as taking all the time amplitude information and mapping it into a single frequency. Here the โmappingโ is multiplying the time signal by a complex exponential at a particular frequency and finding the area under the resulting curve. Hence at each frequency this โmappingโ or, in our case, sum must occur.

What is ๐ exactly?

Much like in the case of sampled time signal we must settle for a sampled frequency signal

๐ ๐ = ๐ ๐ 1

๐ฮ๐ก

1

๐ฮ๐ก can be thought of as the โdistanceโ in frequency between each

successive frequency sample. Compare it to ฮ๐ก for the time sampled signal.

In light of the previous discussion we can say that 1

๐ฮ๐ก comes from

mapping the entire period, ๐ฮ๐ก, of ๐ฅ ๐ก into a single frequency!

๐ ๐ = ๐ฅ ๐ฮ๐ก ๐โ ๐2๐๐๐ T ฮ๐ก

๐โ1

๐=0

So replacing ๐ with ๐

๐ฮ๐ก in our expression we have

Frequency-Domain Housekeeping

With perhaps a slight abuse of notation we might say

๐ฅ ๐ โฑ ๐[๐]

What does it mean?

๐๐ผ = ฮ๐ก ๐ฅ ๐ฮ๐ก ๐ โ ๐2๐๐ผ๐ T

๐โ1

๐=0

At each sampled frequency ๐ = ๐

๐ฮ๐ก (in this case let ๐ = ๐ผ) there is

an amplitude which is found by

The evaluation of ๐๐ at each frequency will populate the frequency domain with an approximation X ๐ known as the Continuous-Time Fourier Transform of ๐ฅ ๐ก .

Give it a name!

๐ ๐ 1

๐ฮ๐ก โ ฮ๐ก โ ๐โฑ๐ฏ ๐ฅ ๐ฮ๐ก

๐โฑ๐ฏ ๐ฅ ๐ฮ๐ก = ๐ฅ ๐ฮ๐ก ๐โ ๐2๐๐๐ T

๐โ1

๐=0

This approximation is known as the Discrete Fourier Transform!

Note: The author of your textbook derives this result differently (Web Appendix H) under the condition that ๐ โช ๐.

Sources

Bracewell, R. (1999). The Fourier Transform and Its Applications (3rd ed.). Boston: McGraw Hill.

Lathi, B. (1992). Linear Systems and Signals (1st ed.). Carmichael, CA: Berkeley-Cambridge Press.

Stewart, J. (2007). Calculus (6th ed.). Belmont, CA: Cengage Learning.