Lecture 27 Frequency Response 2

Transcription

1 Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1

2 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period of ( = 2kHz, rad/sec) In Section 3-4 we showed that periodic waveforms can be represented by a Fourier series of the form The Fourier coefficients are given by the integral 2

3 Application of Ideal Filters Make substitution: If we evaluate the integral, we can show that the complex amplitudes in the spectrum are: Fourier series of the square wave contains only odd harmonics of the fundamental frequency. 3

4 Application of Ideal Filters The spectrum of the square wave with fundamental frequency of 2 khz has (radian) frequency components at rad/sec, rad/sec, rad/sec, etc. The average value of the square wave (over one period) is 4

5 Application of Ideal Filters Now we want to apply an LTI filter to the square wave so that the output will be a sinusoid. The desired spectrum of the output signal consists of two frequency components if we want the output sinusoid at f=2 khz, or rad/sec, the required two complex exponential components will be at khz. 5

6 Application of Ideal Filters The action of the filter operating on the input signal can be described as multiplication of the input spectrum times the frequency response of the filter. That is, if a k represents the Fourier series coefficients of the input square wave, then the output signal has Fourier coefficients The narrow passband of the BPF is centered around the spectrum lines at. The ideal passband can have any nonzero width as long as it includes, but does not include any of the other nonzero spectrum components of the input. Any ideal BPF will give the same output if and 6

7 Application of Ideal Filters The complex amplitudes of the two output spectral lines are found by multiplication, and only the terms of the output Fourier series will be left The output sinusoid 7

8 Application of Ideal Filters The key point of this example is that filtering involves the multiplication of the frequency response times the input spectrum. This provides us with the concept of filters where signal components are either passed through or filtered out of an input signal. 8

9 Time-Domain or Frequency Domain? We have shown that a continuous-time LTI system can be represented in the time domain by its impulse response and in the frequency-domain by its frequency response. Which is to be used in a given situation? When the input signal can be represented as a sum of sinusoids or complex exponentials, then the frequency response method usually provides the simplest solution. If a signal consists of impulses or step functions or other non-sinusoidal signals, convolution of that signal with the impulse response may be the simplest approach. 9

10 Example: Superposition Suppose that an LTI system has impulse response Assume the input is for The input consists of three parts: a constant signal, an impulse, and a cosine wave Because the system is linear, we can take each part separately and use the solution method that is easiest for that input 10

11 Example: Superposition We can treat the constant signal as a complexexponential signal with zero for its frequency. For this we will need to compute the frequency response. Therefore, the output due to the constant signal is 11

12 Example: Superposition The output due to the cosine signal For the impulse component, convolution is easily evaluated by simply shifting and scaling the impulse response, i.e., 12

13 Example: Superposition Overall, the output of the system is 13

14 Lecture 27 Continuous-Time Fourier Transform Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 14

15 Fourier Transform In previous chapters, our presentation of the spectrum was limited to sinusoids and periodic signals. Now we want to develop a completely general definition of the frequency spectrum that will apply to any signal x(t). We will be able to (1) define a precise notion of bandwidth for a signal (2) explain the inner workings of modern communication systems which are able to transmit many signals simultaneously by sharing the available bandwidth (3) define filtering operations that are needed to separate signals in such frequency-shared systems 15

16 Fourier Transform In Chapter 3, we show if a continuous-time signal is periodic with period T 0, then it can represented as a sum of cosines with all frequencies being integer multiples of the fundamental frequency In Chapter 10, we show that a complex-exponential input of frequency produces an output that is also a complex exponential of that same frequency, but modified in amplitude and phase by frequency response of the LTI system. 16

17 Definition of the Fourier Transform Forward Continuous-Time Fourier Transform Inverse Continuous-Time Fourier Transform The forward transform is an analysis integral because it extracts spectrum information The inverse transform is a synthesis integral because it is used to create the time-domain signal from its spectral information. 17

18 Definition of the Fourier Transform Time domain and frequency domain It is common to say that we take the Fourier transform of x(t), meaning that we determine so that we can use the frequency-domain representation of the signal. We often say that we take the inverse Fourier transform to go from the frequency-domain to the time-domain. 18

19 Example: Forward Fourier Transform Consider the one-sided exponential signal Take the Fourier transform of x(t) Time-Domain Frequency-Domain 19 It is rare that we actually evaluate the Fourier transform integral, or the inverse transform integral directly. Instead, the usual route for solving signal processing problems is to use a known Fourier transform pair along with properties of the Fourier transform to get the solution.

20 Example: Unique Inverse Consider the problem of evaluating the following integral: This integral is difficult. However, the integral is a special case of an inverse transform integral, so the uniqueness of the Fourier transform representation guarantees Uniqueness guarantees that there is only one time function that goes with a given Fourier transform. We can do integral by taking the special case of t=-3. 20

21 Fourier Transform It is possible to think of an integral as a sum. We can interpret the synthesis integral as a sum of complex-exponential signals, each having a different complex amplitude if we write the integrand as An integral is a special sum because it is the sum of infinitesimally small quantities. carries the amplitude and phase information for all the frequencies required to synthesize x(t) as a sum of complex-exponential signals. 21

22 Limit of the Fourier Series Recall that the Fourier series representation of a periodic signal are and Now consider a finite-duration signal x(t) that is not periodic, and therefore does not have a Fourier series representation. Nonetheless, we can use x(t) to define one period of a periodic function as long as the period T 0 is longer than the duration of x(t). Start with a finite-duration rectangular pulse 22

23 Limit of the Fourier Series A convenient way to express a periodic signal is to write an infinite sum of time-shifted copies of x(t) T 0 increases relative to T, the periodic signal becomes equal to x(t) over a longer time interval, so we can claim that 23

24 Limit of the Fourier Series Now we examine the Fourier series for each of these periodic signals and take the limit. The fundamental frequency of the periodic signal is, the spectrum lines at integer multiples of. Hence, is the spacing between spectral lines. As, we find that the frequencies contained in the spectrum of become infinitely dense 24

25 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 25

26 Limit of the Fourier Series Similarly, For the examples of Fig. 11-1, the spectra plot 26

27 Limit of the Fourier Series The frequencies get closer and closer together as 27

28 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 28

29 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 29

30 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 30

31 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. 31

32 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. 32

33 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 33

34 Exercise