1. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/1 2.3 Fourier Transform: 2.3.1 From Fourier Series to Fourier Transforms  First, the signal x(t) must satisfy the following condition: 1. x(t) is absolutely integrable on the real line; that is, 2. x(t) is an energy type signal Then the Fourier transform (or Fourier integral) of x(t), defined by The original signal can be obtained from its Fourier transform by

2. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/2 FOURIER TRANSFORM The following observations concerning the Fourier transform X(f) is generally a complex function  Its magnitude | X(f) | and phase  X(f) represent the amplitude and phase of various frequency components in x(t)  X(f), | X(f) |, | X(f) | 2 is sometimes referred to as the spectrum of the signal x(t) Shorthand for both FT and iFT relations:  X(f) : Fourier transform of x(t), the notation:  x(t) : Inverse Fourier transform of X(f), the notation: If the variable in the Fourier transform is ω rather than f, then

3. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/3 FOURIER TRANSFORM Proof of FT  From FS:  Let

4. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/4 Example 2.3.1: Example 2.3.2: Example 2.3.3:

5. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/5 Fourier Transform of Real, Even, and Odd Signals For real x(t), the transform X(f) is a Hermitian function: If, in addition to being real, x(t) is an even signal : the Fourier transform X(f) will be real and even. If x(t) is real and odd : the real part of its Fourier transform vanishes and X( f ) will be imaginary and odd. Figure 2.36 Magnitude and phase of the spectrum of a real signal.

6. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/6 Signal Bandwidth The bandwidth of a signal represents the range of frequencies present in the signal The higher the bandwidth, the larger the variations in the frequencies present In general, we define the bandwidth of a real signal x(t) as the range of positive frequencies present in the signal In order to find the bandwidth of x(t)  we first find X(f), which is the Fourier transform of x(t)  we find the range of positive frequencies that X(f) occupies The bandwidth is BW = W max – W min  W max : Highest positive frequency present in X(f)  W min : Lowest positive frequency present in X(f)

7. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/7 2.3.2 Basic Properties of the Fourier Transform Theorem : Linearity Theorem : Duality

8. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/8 Example 2.3.5: Ex 2.3.1 에서 이므로 Example: Ex 2.3.2 에서 이므로 Example 2.3.6: Ex 2.3.3 에서 이므로 Example 2.3.4:

9. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/9 Basic Properties of the Fourier Transform Theorem : Shift in Time Domain Theorem : Scaling If a > 1, then x(at) is a contracted form of x(t) If a < 1, x(at) is an expanded version of x(t)  If we expand a signal in the time domain, its frequency-domain representation (Fourier transform) contracts  If we contract a signal in the time domain, its frequency domain representation expands  This is exactly what we expect since contracting a signal in the time domain makes the changes in the signal more abrupt, thus increasing its frequency content.

10. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/10 Basic Properties of the Fourier Transform Theorem : Convolution  This theorem is very important and is a direct result of the fact that the complex exponentials are eigenfunctions of LTI systems (or, equivalently, eigenfunctions of the convolution operation).  Finding the response of an LTI system to a given input is much easier in the frequency domain than it is the time domain.  This theorem is the basis of the frequency-domain analysis of LTI systems.

11. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/11 Example 2.3.9: Example 2.3.10: Example 2.3.11:

12. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/12 Basic Properties of the Fourier Transform Theorem : Modulation This theorem is the dual of the time-shift theorem. The time-shift theorem says that a shift in the time domain results in a multiplication by a complex exponential in the frequency domain The modulation theorem states that a multiplication in the time domain by a complex exponential results in a shift in the frequency domain A shift in the frequency domain is usually called modulation

13. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/13 Example 2.3.14 Determine the Fourier transform of the signal Solution In Chapter 3, we will see that this relation is the basis of the operation of amplitude modulation systems. Figure 2.38 Effect of modulation in both the time and frequency domain.

14. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/14 Basic Properties of the Fourier Transform Theorem : Parseval’s Relation  If the Fourier transforms of the signals x(t) and y(t) are denoted by X(f) and Y(f) respectively, then  Note that if we let y(t) = x(t), we obtain This is known as Rayleigh's theorem and is similar to Parseval's relation for periodic signals

15. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/15 Example 2.3.16:

16. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/16 Basic Properties of the Fourier Transform Theorem : Time Autocorrelation The time autocorrelation function of the signal x(t) is denoted by R x (  ) and is defined by The autocorrelation theorem states that The Fourier transform of the autocorrelation of a signal is always a real-valued, positive function

17. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/17 Basic Properties of the Fourier Transform Theorem : Differentiation Theorem : Differentiation in the Frequency Domain

18. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/18 Example 2.3.17: Example 2.3.18: 기출문제 :

19. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/19 Basic Properties of the Fourier Transform Theorem : Integration Theorem : Moments For the special case of n = 0, we obtain this simple relation for finding the area under a signal, i.e.,

20. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/20 Fourier Transform Pairs

21. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/21 Fourier Transform Properties

22. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/22 2.3.4 Transmission over LTI Systems The output of an LTI system is equal to the convolution of the input and the impulse response of the system If we translate this relationship in the frequency domain using the convolution theorem, then X(f), H(f), and Y(f) are the Fourier transforms of the input, system impulse response, and the output, respectively. The input-output relation for an LTI system in the frequency domain is much simpler than the corresponding relation in the time domain In the time domain, we have the convolution integral; however, in the frequency domain we have simple multiplication To find the output of an LTI system for a given input, we must find the Fourier transform of the input and the Fourier transform of the system impulse response. Then, we must multiply them to obtain the Fourier transform of the output. To get the time-domain representation of the output, we find the inverse Fourier transform of the result

23. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/23 Transmission over LTI Systems Lowpass signals Signals with a frequency domain representation that contains frequencies around the zero frequency and does not contain frequencies beyond some W l Ideal lowpass filter An LTI system that can pass all frequencies less than some W and rejects all frequencies beyond W An ideal lowpass filter will have a frequency response that is 1 for all frequencies -W  f  W and is 0 outside this interval W is the bandwidth of the filter.

24. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/24 Transmission over LTI Systems Ideal highpass filter There is unity outside the interval -W  f  W and zero inside Ideal bandpass filters have a frequency response that is unity in some interval W 1  |f|  W 2 and zero otherwise The bandwidth of the filter is W 2 – W 1 Figure 2.45 Various filter types.

25. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/25 2.6 HILBERT TRANSFORM AND ITS PROPERTIES In this section, we will introduce the Hilbert transform of a signal and explore some of its properties The Hilbert transform is unlike many other transforms because it does not involve a change of domain  In contrast, Fourier, Laplace, and z-transforms start from the time-domain representation of a signal and introduce the transform as an equivalent frequency-domain (or more precisely, transform-domain) representation of the signal The resulting two signals are equivalent representations of the same signal in terms of two different arguments, time and frequency Strictly speaking, the Hilbert transform is not a transform in this sense  First, the result of a Hilbert transform is not equivalent to the original signal, rather it is a completely different signal  Second, the Hilbert transform does not involve a domain change, i.e., the Hilbert transform of a signal x(t) is another signal denoted by in the same domain (i.e., time domain with the same argument t)

26. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/26 HILBERT TRANSFORM The Hilbert transform of a signal x(t) is a signal whose absolute frequency components lag the frequency components of x(t) by 90   has exactly the same frequency components present in x(t) with the same amplitude–except there is a 90  phase delay  The Hilbert transform of x(t) = Acos(2  f 0 t +  ) is Acos(2  f 0 t +  - 90  ) = Asin(2  f 0 t +  ) A lag of  /2 at all absolute frequencies  e j2  f 0 t will become  e -j2  f 0 t will become At positive frequencies, the spectrum of the signal is multiplied by -j At negative frequencies, it is multiplied by +j  This is equivalent to saying that the spectrum (Fourier transform) of the signal is multiplied by -jsgn(f).

27. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/27 HILBERT TRANSFORM  In this section, assume that x(t) is real The operation of the Hilbert transform is equivalent to a convolution, i.e., filtering

28. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/28 Example 2.6.1 Determine the Hilbert transform of the signal x(t) = 2sinc(2t) Solution We use the frequency-domain approach to solve this problem. Using the scaling property of the Fourier transform, we have In this expression, the first term contains all the negative frequencies and the second term contains all the positive frequencies To obtain the frequency-domain representation of the Hilbert transform of x(t), we use the relation which results in Taking the inverse Fourier transform, we have

29. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/29 HILBERT TRANSFORM AND ITS PROPERTIES Evenness and Oddness The Hilbert transform of an even signal is odd, and the Hilbert transform of an odd signal is even  Proof If x(t) is even, then X(f) is a real and even function Therefore, -jsgn(f)X(f) is an imaginary and odd function Hence, its inverse Fourier transform will be odd If x(t) is odd, then X(f) is imaginary and odd Thus -jsgn(f)X(f) is real and even Therefore, is even Sign Reversal Applying the Hilbert-transform operation to a signal twice causes a sign reversal of the signal, i.e.,  Proof

30. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/30 HILBERT TRANSFORM AND ITS PROPERTIES Energy The energy content of a signal is equal to the energy content of its Hilbert transform  Proof Using Rayleigh's theorem of the Fourier transform, we have Orthogonality The signal x(t) and its Hilbert transform are orthogonal  Proof Using Parseval's theorem of the Fourier transform, we obtain In the last step, we have used the fact that X(f) is Hermitian; | X(f)| 2 is even

31. Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/31 Recommended Problems Textbook Problems from p113 2.39.3, 2.39.5, 2.39.9, 2.39.11 2.41.1-4 2.43.(b) 2.46.1, 2.46.4, 2.46.6, 2.46.7, 2.46.9 2.47 2.49.(a), 2.49.(d), 2.49.(e) 2.50 2.53.4 2.59.3, 2.59.6 2.60, 2.62, 2.63, 2.65