SIGNALS & SYSTEMS (ENT 281) Chapter 4: Fourier Transform Dr. Hasimah Ali School of Mechatronic Engineering University Malaysia Perlis

Applies for both periodic and nonperiodic (aperiodic) signals. 3.1 INTRODUCTION Fourier transform is a transformation technique which transforms signals from the continuous time domain to the corresponding frequency domain ot vice versa. Applies for both periodic and nonperiodic (aperiodic) signals. The Fourier transform derived in this chapter is called Continuous-time Fourier transform (CTFS) Applications: Analysis of linear time-invariant (LTI) systems, cryptography, signal analysis, signal processing, astronomy, RADAR and etc.

Fourier Transform or the Fourier integral of x(t) defined as: X(ω) represents the frequency spectrum of x(t) and is called the spectral density function.

The inverse Fourier Transform of X(ω): Fourier transform pair can be denoted as: or

3.3 MAGNITUDE AND PHASE REPRESENTATION OF FOURIER TRANSFORM (FT) The magnitude and phase representation of FT is the tool used to analyse the transformed signal. In general X(ω) is a complex valued function of ω. Therefore, X(ω) can be written as: XR(ω): is the real part of X(ω). XL(ω):is the imaginary part of X(ω).

3.3 MAGNITUDE AND PHASE REPRESENTATION OF FOURIER TRANSFORM (FT) The magnitude of X(ω): The phase of X(ω): The plot of versus is known as amplitude spectrum. The plot of versus is known as phase spectrum. The amplitude spectrum and phase spectrum together is called frequency spectrum.

3.4 EXISTENCE OF FOURIER TRANSFORMS (FT) The conditions for a function x(t) to have Fourier transform, called Dirichlet’s condition, are: x(t) is absolutely integrable over the interval to ,that is: X(t) has a finite number of discontinuities in every finite time interval. Further, each of these discontinuities must be finite. X(t) has a finite number of maxima and minima in every finite time interval

3.5 FOURIER TRANSFORMS OF STANDARD SIGNALS 1. Impulse Function Given, Then, or

Hence, the Fourier transform of a unit impulse function is unity.

2. One-sided Real Exponential Signal

2. One-sided Real Exponential Signal or

3. Double-sided Real Exponential Signal

Cont.….

Cont. … or for all ω for all ω

4. Complex Exponential Signal To find the FT of complex exponential function, consider finding the inverse FT of . Let

5. Constant Amplitude (1) Since x(t) = 1 is not absolutely integrable, we cannot find its FT directly. So the FT of x(t) = 1 is determined through inverse Fourier transform of Consider

Cont. …

6. Signum Function sgn (t)

Cont. …

Cont. …

6. Unit Step Function u(t)

Cont. …

Cont. …

6. Rectangular pulse rect (t)

6. Rectangular pulse rect (t)

6. Rectangular pulse rect (t)

6. Cosine Wave

6. Sine Wave

Properties of CT Fourier Transform 1. Linearity: Then: where a and b are arbitrary constants

Example1: Suppose we want to find the Fourier transform (FT) of cos ω0t. The cosine signal can be written as a sum of two exponentials as follows: The linearity property of the FT: Similarly, the Fourier transform of sin ω0t is:

Properties of CT Fourier Transform ‘*’ Complex Conjugate 2. Symmetry: If x(t) is a real-valued time signal, then If X(ω) in the polar form: Taking the complex conjugate both sides i.e., the magnitude spectrum is an even function of frequency and the phase spectrum is an odd function of frequency Replacing each ω by - ω The left-hand sides of the last 2 equations are equal

Example2: Consider an even and real-valued signal x(t). Its transform X(ω)is: Since x(t)cos ω t is an even function and x(t) sin is an odd function of t, we have:

Properties of CT Fourier Transform 3. Time Shifting (Delay): If: Then: Similarly:

Properties of CT Fourier Transform 4. Time Scaling: If: Then: where: is real constant Expanded in time Compressed in time

Example3: Suppose we want to determine the Fourier transform of the pulse The Fourier transform of is,

Properties of CT Fourier Transform 5. Time-Differentiation: If: Then: 6.Time-Integration:

The first term has πδ(ω) Example4: Consider the unit-step function. The function can be written: The first term has πδ(ω) The derivatives of discontinuous signals in terms of the delta function

Properties of CT Fourier Transform 7. Convolution: If: and: Then:

Example5: Consider an LTI system with impulse response h(t). Compute the convolution using the convolution property of the FT whose input is the unit step function u(t). Solution: Taking the inverse FT:

Properties of CT Fourier Transform 8. Duality: If: Then:

Properties of CT Fourier Transform 9. Modulation: If: Convolution in the frequency domain is carried out exactly like convolution in the domain. That is, Then:

Cont,. … The importance of this property is that the spectrum of a signal such as x(t) cos ω0t can be easily computed. Since, It follow that

Example6: Consider the signal: Where p(t) is the periodic impulse train with equal-strength impulses. Analytically, p(t) can be written as: Solution: Using the sampling property of the delta function, we obtain:

Cont. … The periodic of the periodic impulse train p(t) it itself a periodic impulse train; Specifically: By modulation property,

Example5: Consider an LTI system with impulse response h(t). Compute the convolution using the convolution property of the FT whose input is the unit step function u(t). Solution: Taking the inverse FT:

Example6:

Applications of the Fourier Transform Amplitude Modulation: The goal of communication system: to convey the information from one point to another. The signal need to be converted into useful form before sending the information through the transmission channel which is known as modulation. Reasons for conversion: 1) To transmit information efficiently, 2)to overcome hardware limitation. 3)to reduce noise and interference, 4) to utilize the EM spectrum efficiently. Modulation is the process of merging two signals to form a third signal with desirable characteristics of both.

Cont. … Consider the signal multiplier. y(t) m(t) x(t) X The output is the product of information-carrying signal x(t) and the signal m(t) as carrier signal. This scheme is known as amplitude modulation, which has many forms depending on m(t). We concentrate only on the case m(t) = cos ω0t , which represent the practical form of modulation and is referred as double-sideband (DSB) amplitude modulation.

Cont. … The output of the multiplier is. Use the property of convolution in the frequency domain to obtain its spectrum

Cont. … The magnitude spectra of x(t) and y(t) given by: The part of the spectrum of Y(ω) centered at + ω0 is the result of convolving X(ω) with δ(ω-ω0 ) , And the part centered at –ω0 is the result of convolving X(ω) with δ(ω+ω0 ). This process of shifting the spectrum of the signal by is necessary because low-frequency (baseband) information signal cannot be propagated easily by radio waves.

Cont. … The process of extracting the information signal from the modulated signal is called demodulation. Synchronous demodulation =>used to perform amplitude demodulation. The output of the multiplier is, Hence,

Magnitude Spectrum of z(t) The low-pass-filter frequency spectrum Cont. … To extract the original information signal x(t), the signal z(t) is passed through the system with frequency response H(ω). Magnitude Spectrum of z(t) The low-pass-filter frequency spectrum

Cont. … Such a system is referred to as a low-pass filter, since it passes only low-frequency components of the input signal and filters out all frequency higher than ωB, the cutoff frequency of the filter. b)The low-pass-filter frequency spectrum, c) The extracted information Note that if , and there were no transmission losses involved, then the energy of final signal is one-fourth that of the original signal because the total demodulated signal contains energy located at ω= 2ω0, that is eventually discarded by the receiver.

Example: A signal x(t) = cos 10t + 2 cos 20t modulates the carrier signal xc(t) = cos 100t. Find the spectrum of the modulated signal.

REFERENCES A. Anand Kumar, “Signals & Systems”, PHI Learning Private Limited. 2nd edition. New Delhi. Simon Haykin and Barry Van Veen, “Signals and Systems”, Wiley, 2nd Edition, 2002 M.J. Roberts, “Signals and Systems”, International Edition, McGraw Hill, 2nd Edition 2012