Lecture 28 Continuous-Time Fourier Transform 2
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1 Lecture 28 Continuous-Time Fourier Transform 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/14 1
2 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 points on the frequency axis that approaches the continuous variable As a result, we can claim that 2
3 Limit of the Fourier Series Similarly, For the examples of Fig. 11-1, the spectra plot 3
4 Limit of the Fourier Series The frequencies get closer and closer together as 4
5 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 transform exists or not check the magnitude of 5
6 Existence and Convergence To obtain a sufficient condition for existence of the Fourier transform 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 6
7 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 signal because it is nonzero only on the right side. Time-Domain Frequency-Domain 7
8 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. Thus, the right-sided exponential signal is guaranteed to have a Fourier transform if it dies out with increasing t, which requires a > 0. 8
9 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 as functions of frequency. 9
10 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 quickly. In the frequency-domain, as a increases, the Fourier transform spreads out Signals that are short in time duration are spread out in frequency 10
11 Exercise
12 Rectangular Pulse Signals Consider the rectangular pulse The Fourier transform is Time-Domain Frequency-Domain 12
13 Rectangular Pulse Signals The Fourier transform of the rectangular pulse signal is called a sinc function. The formal definition of a sinc function is Time-Domain Frequency-Domain 13
14 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 Note that we could also use the small angle approximation for the sine function to obtain the same result 14
15 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 follows that for or 15
16 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 real even-symmetric Fourier transform. 16
17 Bandlimited Signals 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 bandlimited Fourier transform 17
18 Bandlimited Signals We want to determine the time-domain signal that has this Fourier transform, i.e., we need to evaluate the inverse transform integral 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 nonzero multiplies of 18
19 Bandlimited Signals 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. Time-Domain Frequency-Domain 19
20 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. Time-Domain Frequency-Domain 20
21 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 Time-Domain Frequency-Domain 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 21
22 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 impulse at,. By substituting into the inverse transform integral Time-Domain Frequency-Domain 22
23 Sinusoids Time-Domain Frequency-Domain The result is not unexpected. It says that a complexexponential signal of frequency has a Fourier transform that is nonzero only at the frequency. The result is the basis for including all periodic functions in our Fourier transform framework. Consider the signal 23
24 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 Thus, the Fourier transform of the real sinusoid x(t) is 24
25 Sinusoids So we have the Fourier transform pair Time-Domain Frequency-Domain Note that the size (area) of the impulse at negative frequency is the complex conjugate of the size of the impulse at the positive frequency. 25
26 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 where and 26
27 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. 27
28 Periodic Signals Time-Domain Frequency-Domain 28 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.
29 Example: Square Wave Transform A periodic square wave where T 0 = 2T We also obtain the DC coefficient by evaluating the integral 29
30 Example: Square Wave Transform After substituting, we obtain If we substitute this into (11.35) we obtain the equation for the Fourier transform of a periodic square wave: 30
31 Example: Square Wave Transform This figure shows the Fourier transform of the square wave for the case T 0 = 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 sizes dictated by the a k coefficients. 31
32 Example: Transform of Impulse Train Consider the periodic impulse train Express it as a Fourier series To determine the Fourier coefficients {a k }, we must evaluate Fourier series integral over one convenient period The Fourier coefficients for the periodic impulse train are all the same size. 32
33 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 a k into the general expression for, we obtain Therefore, the Fourier transform of a periodic impulse train is also a periodic impulse train. 33
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