Lecture 28 Continuous-Time Fourier Transform 2

Fundamentals of Digital Signal Processing

Lecture 28 Continuous-Time Fourier Transform 2

Fundamentals of Digital Signal ProcessingSpring, 2012

Wei-Ta Chu2012/6/14

1 DSP, CSIE, CCU

Limit of the Fourier Series� Rewrite (11.9) and (11.10) as

� As , the fundamental frequency gets very small and the set defines a very dense set of

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small and the set defines a very dense set of points on the frequency axis that approaches the continuous variable

� As a result, we can claim that

Limit of the Fourier Series� Similarly,

� For the examples of Fig. 11-1, the spectra plot

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the spectra plot

Limit of the Fourier Series� The frequencies get closer and closer together as

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Existence and Convergence� The Fourier transform and its inverse are integrals with

infinite limits.

� An infinite sum of even infinitesimally small quantities might not converge to a finite result.

� To aid in our use of the Fourier transform it would be helpful to be able to determine whether the Fourier

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helpful to be able to determine whether the Fourier transform exists or not

check the magnitude of

Existence and Convergence� To obtain a sufficient condition for existence of the

Fourier transform

The last step follows that for all t and

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The last step follows that for all t and

� Thus, a sufficient condition for the existence of the Fourier transform ( ) is

Sufficient Condition for Existence of

Right-Sided Real Exponential Signals� Fourier transform can represent non-periodic signals in

much the same way that the Fourier series represents periodic signals

� The signal is a right-sided exponential signalbecause it is nonzero only on the

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because it is nonzero only on the right side.

Time-Domain Frequency-Domain

Right-Sided Real Exponential Signals� Substitute the function into (11.15) we

obtain

� This result will be finite only if at the upper limit of is bounded, which is true only if a > 0.

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of is bounded, which is true only if a > 0.

� Thus, the right-sided exponential signal is guaranteed to have a Fourier transform if it dies out with increasing t, which requires a > 0.

Right-Sided Real Exponential Signals� The Fourier transform is a

complex function of .

� We can plot the real and imaginary parts versus , or plot the magnitude and phase angle asfunctions of frequency.

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functions of frequency.

Bandwidth and Decay Rate� These figures show a fundamental property of Fourier

transform representations – the inverse relation between time and frequency.

� a controls the rate of decay

� In the time-domain, as a increases, the exponential dies out more

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the exponential dies out more quickly.

� In the frequency-domain, as aincreases, the Fourier transformspreads out

� Signals that are short in time duration are spread out in frequency

Exercise 11.2

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Rectangular Pulse Signals� Consider the rectangular pulse

� The Fourier transform is

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Rectangular Pulse Signals� The Fourier transform of the rectangular pulse signal is

called a sinc function.

� The formal definition of a sinc function is

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Rectangular Pulse Signals� Properties of the sinc function

� 1. The value at is . When we attempt to evaluate the sinc formula at , we obtain . However, using L’Hopital’s rule from calculus, we obtain

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� Note that we could also use the small angle approximation for the sine function to obtain the same result

Rectangular Pulse Signals� 2. The zeros of the sinc function are at nonzero integer

multiples of , where T is the total duration of the pulse.

� It crosses zero at regular intervals because we have in the numerator.

� Since for where n is an integer, it

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� Since for where n is an integer, it follows that for or

Rectangular Pulse Signals� 3. Because of the in the denominator of , the

function dies out with increasing , but only as fast as

� 4. is an even function, i.e.,

Thus the real even-symmetric rectangular pulse has a

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Thus the real even-symmetric rectangular pulse has a real even-symmetric Fourier transform.

BandlimitedSignals� We define a bandlimited signal as one whose Fourier

transform satisfies the condition for with

� The frequency is called the bandwidth of the bandlimited signal.

� One ideally bandlimitedFourier transform

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� One ideally bandlimitedFourier transform

BandlimitedSignals� We want to determine the time-domain signal that has

this Fourier transform, i.e., we need to evaluate the inverse transform integral

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� It has the form of a sinc function

� This signal has a peak value of at t = 0, and the zero

crossings are spaced at nonzeromultiplies of

BandlimitedSignals� Note the inverse relationship between time width and

frequency width.

� If we increase , the bandwidth is greater, but the first zero crossing in the time domain moves closer to t= 0 so the time-width is smaller.

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Impulse in Time or Frequency� The impulse time-domain signal is the most

concentrated time signal that we can have. Therefore, we might expect that its Fourier transform will have a very wide bandwidth, and it does. The Fourier transform of contains all frequencies in equal amounts.

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amounts.


Impulse in Time or Frequency� Likewise, we can examine an impulse in frequency, if

we define the Fourier transform of a signal to be

� We can show by substitution into (11.2) that x(t) = 1 for all t and thereby obtain the Fourier transform pair

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� The constant signal x(t) = 1 for all t has only one frequency, namely DC, and we see that its transform is an impulse concentrated at


Sinusoids� We will show how to determine the Fourier transform

of a periodic signal. We know that periodic signals can be represented as Fourier series. However, there are distinct advantages for bring this class of signals under the general Fourier transform umbrella.

� Suppose that the Fourier transform of a signal is an

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� Suppose that the Fourier transform of a signal is an impulse at , . By substituting into the inverse transform integral


Sinusoids

� The result is not unexpected. It says that a complex-exponential signal of frequency has a Fourier transform that is nonzero only at the frequency . The result is the basis for including all periodic


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The result is the basis for including all periodic functions in our Fourier transform framework.

� Consider the signal

Sinusoids

� Since integration is linear, it follows that the Fourier transform of a sum of two or more signals is the sum of their corresponding Fourier transforms.


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� Thus, the Fourier transform of the real sinusoid x(t) is


Sinusoids

� So we have the Fourier transform pair


Note that the size (area) of the

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Note that the size (area) of the impulse at negative frequencyis the complex conjugate of the size of the impulse at the positive frequency.

Periodic Signals� Now we are ready to obtain a general formula for the

Fourier transform of any periodic function for which a Fourier series exists.

� A periodic signal can be represented by the sum of complex exponentials

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where and

Periodic Signals� The Fourier transform of a sum is the sum of

corresponding Fourier transforms

� Thus, any periodic signal with fundamental frequency is represented by the following Fourier transform pair as this figure.

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as this figure.

Periodic Signals


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The key ingredient is the impulse function which allows us to define Fourier transforms that are zero at all but a discrete set of frequencies.

Example: Square Wave Transform� A periodic square wave where T0 = 2T

� We also obtain the DC coefficient by evaluating the integral

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integral

Example: Square Wave Transform

� After substituting , we obtain

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� If we substitute this into (11.35) we obtain the equation for the Fourier transform of a periodic square wave:

Example: Square Wave Transform� This figure shows the Fourier transform of the square

wave for the case T0 = 2T. The Fourier coefficients are zero for even multiples of , so there are no impulses at those frequencies. Any periodic signal with fundamental frequency will have a transform with impulses at integer multiples of , but with different

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impulses at integer multiples of , but with different sizes dictated by the ak coefficients.

Example: Transform of Impulse Train� Consider the periodic impulse train

� Express it as a Fourier series

� To determine the Fourier coefficients {ak}, we must evaluate Fourier series integral over one convenient

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evaluate Fourier series integral over one convenient period

� The Fourier coefficients for the periodic impulse train are all the same size.

Example: Transform of Impulse Train� The Fourier transform of a periodic signal represented

by a Fourier series as in (11.42) is of the form

� Substituting ak into the general expression for , we obtain

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� Therefore, the Fourier transformof a periodic impulse train is also a periodic impulse train.

Lecture 28 Continuous-Time Fourier Transform 2

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Transcript of Lecture 28 Continuous-Time Fourier Transform 2