Fundamentals of Digital Signal Processing Lecture 28 Continuous-Time Fourier Transform 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/14 1 DSP, CSIE, CCU
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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Time-Domain Frequency-Domain
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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Time-Domain Frequency-Domain
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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Time-Domain Frequency-Domain
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.
Time-Domain Frequency-Domain
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
Time-Domain Frequency-Domain
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
Time-Domain Frequency-Domain
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
Time-Domain Frequency-Domain
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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.
Time-Domain Frequency-Domain
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� Thus, the Fourier transform of the real sinusoid x(t) is
Time-Domain Frequency-Domain
Sinusoids
� So we have the Fourier transform pair
Time-Domain Frequency-Domain
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
Time-Domain Frequency-Domain
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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.