Engineering in EEE 303: Signals and Linear Systems

The Fourier Transform

Let be a nonperiodic signal of finite duration, i.e.,

Let us form a periodic signal by extending to as,

,

[i.e., the period is infinity]

Then,

(01)Or,

Let us now define as,

Thus,

.

Substituting this in equation (01) we get,

As , . Let us assume.

Thus,

Or,

(02)

in equation (02) is called the Fourier Integral. Thus a finite duration signal is represented by Fourier integral instead of Fourier series.

The function is called the Fourier transform of.

Symbolically these two pairs are represented as,

And

Alternatively,

.Fourier Spectra

The Fourier transform is, in general, complex, and it can be expressed as,

If is real,

Thus,

Therefore, we can conclude that for real signal, the amplitude spectrum is an even function and the phase spectrum is an odd function of .

The condition for the convergence of Fourier transform is the same as the Fourier series.

Example1. Find the Fourier transform of

.2. Find the Fourier transform of .

3. Find the inverse Fourier transform of .

.

Thus, or,

.

4. Find the inverse Fourier transform of .

Thus,

We know, ; Thus,

5. Find the Fourier transform of the rectangular pulse shown in Figure.

The magnitude spectrum is, , and the phase spectrum is, .6. Find the inverse Fourier transform of the rectangular spectrum shown below.

. The plot is shown in Figure above.Some Properties of Fourier Transform

1. Symmetry property: If . (duality property)

Example: Apply symmetry property to show that .

2. Scaling Property: If .

3. Time-shifting Property: If .4. Frequency-shifting Property: If .

Example: Find the Fourier transform of the gate pulse shown in Figure below.

We get the Fourier transform by applying time-delay property to the F.T. of rectangular pulse (symmetrical).

Thus,

.

Example: Sketch the Fourier transform of using frequency shifting property. [property 4] . Therefore, . The sketch is shown in Figure below. Here, .

5. Time and Frequency convolution: .

6. Time differentiation and time integration: .

(a)

Therefore, =,

or, . (b)

Using convolution property,

Therefore, .Example: Using the time-differentiation property, find the F.T. of the triangle illustrated in figure below.

Performing F.T. of the first equation,

EMBED Equation.DSMT4

Fourier transform of periodic signalLet,

(periodic). Performing L.T. we get,

.Therefore, the Fourier transform of a periodic signal is a train of impulses occurring at harmonically related frequencies where the impulse strength is, .For example, consider the Fourier series and Fourier transform of the signal. Energy of aperiodic signal

Energy of aperiodic signal,

Or,

i.e., . This relation is called the Parsivals relation for aperiodic signal.

The term is called the energy spectral density of the signal. If it is integrated over all the frequencies and multiplied by we get the energy of the aperiodic signal.

Example: The signals shown in Figure below are modulated signal with carrier. Find the Fourier transform of these signals.

(a) Here a triangular pulse of width is modulated with a carrier . Now the Fourier transform of triangular pulse is, .

And the F.T. of carrier is,

Therefore,

Or,

[ans]

(b) This is the time-shifted version of (a). Here the time-shift is . Time shift corresponds to phase shift of in frequency domain.The result is, .(c) Here cosine wave is multiplied by a rectangular pulse of width with a time-shift of .

Here the F.T. of rectangular pulse is (symmetric w.r.t. y-axis), . is the same as in (a). Thus,

EMBED Equation.DSMT4 Hence, . [ans]Signal transmission through LTI systemWe know, . Using convolution property, .Therefore, .

Remember that, is called the frequency response of a system. In order to find the response of a system using F.T. we have to get first. Then we will get using inverse Fourier transform of . Example

Find the response of a system shown below when an input of the system is.

Performing F.T. on both side,

. But,

Performing inverse Fourier transform, [ans]If , then the characteristic mode of input and output are the same and resonance will occur. If we put the condition in , it will become indeterminate. We can find using LHospital rule.Hence,

.System response to periodic inputLet, . Performing F.T. we get, .Now, .

Performing inverse F.T. we get,

Therefore, the output will be also periodic. The amplitude of the nth harmonic components will be, .

Some More Examples1. Determine the Fourier transform of the complex sinusoidal pulse, .The function z(t) may be expressed as, where, .

In Fourier transformed domain, . [frequency shifting property]

Now, . Here . Thus, .Hence,

[Ans]

2. Show that the differentiation in frequency corresponds to multiplication in time by -jt.

Therefore,

Hence the statement.

3. Use the frequency differentiation property to find the F.T. of .

We know,

Therefore,

Or, ; or, [Ans]4. Let the input to a system with impulse response be . Find the output of the system .

. Now and .

. Performing inverse Fourier transform,

[Ans]Windowing operationThe process of truncating a function is called windowing. It is represented mathematically by multiplying the signal, by a window function, . If is the windowed signal, .In frequency domain it can be viewed as, . The windowing operation and its effect in frequency domain is shown in figure below.

The general effect of window is to smooth detail in and introduce oscillation near discontinuities in. The smoothing is the consequence of the main lobe of width (for rectangular window) while the oscillations are due to the oscillations in the side lobes of .Fourier Transform of

1. ; We know that, and

Thus, .

[Ans]

2.

Therefore,

EMBED Equation.DSMT4

[Ans]Example

The output of a system in response to an input is. Find the frequency response and the impulse response of the system.

,

EMBED Equation.DSMT4 .

EMBED Equation.DSMT4

Performing inverse F.T. we get, .

[Ans]

Find the frequency response and the impulse response of the system .Signal distortion during transmission

For distortionless transmission through an LTI system we require that the exact input signal shape be reproduced at the output although its amplitude may be different and it may be delayed in time. Therefore,

.Thus for distortionless transmission the system must have, .

and .

i.e., the amplitude of must be constant over the entire frequency range and the phase of must be linear with frequency.

A system may have flat amplitude response, but it will be distorted if system is not constant.Application of Fourier transform

1. Modulation and demodulation

To allow different frequency band for different channel

For effective radiation of signal antenna size must be of the order of the wavelength of the signal to be radiated. Shifting the signal to the higher frequency by modulation solves the problem.

Figure left shows the process of amplitude modulation with a sinusoidal carrier. We choose for convenience.

The original signal can be recovered by modulating with the same sinusoidal carrier and applying low-pass filter to the result.

Or,

EMBED Equation.DSMT4 If modulator and demodulator are not synchronized,

EMBED Equation.DSMT4 .In this case, , the output will be zero. For maximum output signal the oscillators must be in phase. This requires careful synchronization between modulator and demodulator._1262103233.unknown

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