1. ELEN 4610: Analog Communications Chapter #2: Fourier Representation of Signals and Systems Prof. Caroline GonzálezMatlab and Simulink Tutorialhttp://www.mathworks.com/academia/student_center/tutorials Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007. Ch 2-1 In this chapter, we will study: Definition of the Fourier Transform Properties of the Fourier Transform The Inverse Relationship between Time and Frequency Dirac Delta Function Fourier Transform of Periodic Signals Power Spectral Density Haykin, S., and M. Moher, Introduction to Analog & Digital 2 Communications, 2nd ed., Wiley, 2007. 1
2. The Fourier Transform (FT) The FT relates the frequency-domain description of a signal to its time-domain description. − Determine the frequency content of a continuous-time signal. − Evaluates what happens to this frequency content when the signal is passed through a linear time-invariant (LTI) system. − A signal can only be strictly limited in the time domain or the frequency domain, but not both. − Bandwidth is an important parameter in describing the spectral content of a signal and the frequency response of a LTI filter. Haykin, S., and M. Moher, Introduction to Analog & Digital 3 Communications, 2nd ed., Wiley, 2007.Definition of the FT Advantages of using frequency-domain analysis − Resolution into eternal sinusoids presents the behavior as the superposition of steady-state effects. − Usually the time-domain analysis involves solving differential equations, but in the frequency domain involves simple algebra equations. − Provides the frequency content of a signal. Haykin, S., and M. Moher, Introduction to Analog & Digital 4 Communications, 2nd ed., Wiley, 2007. 2
3. Dirichlet’s Conditions For the FT of a signal g(t) to exist, it is sufficient, but not necessary, that g(t) satisfies: − The function g(t) is single-valued, with a finite number of maxima and minima in any finite time interval. − The function g(t) has a finite number of discontinuities in any finite time interval. − The function g(t) is absolutely integrable or the g(t) is an energy-like signal. ∞ ∫ g (t )dt < ∞ −∞ ∞ 2 ∫ g (t ) −∞ dt < ∞ 5 Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.Continuous Spectrum The FT is a complex function of frequency so that G ( f ) = G ( f ) e jθ ( f ) where G ( f ) is the continuous amplitude spectrum θ ( f ) is the continuous phase spectrum For a real-value function g(t) the FT has the following characteristics G ( − f ) = G* ( f ) G (− f ) = G ( f ) θ ( − f ) = −θ ( f ) 6 Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007. 3
4. Continuous Spectrum In conclusion − The spectrum of a real-valued signal exhibits conjugate symmetry. The amplitude spectrum of a signal is an even function of the frequency; the amplitude spectrum is symmetric with respect to the origin f=0. The phase spectrum of a signal is an odd function of the frequency; the phase spectrum is antisymmetric with respect to the origin f=0 Haykin, S., and M. Moher, Introduction to Analog & Digital 7 Communications, 2nd ed., Wiley, 2007.Examples Rectangular Pulse (Example 2.1) − Matlab Demo Decaying Exponential Pulse (Ex. 2) − Matlab Demo Rising Exponential Pulse (Ex. 2) − Matlab Demo Drill P2.1 Haykin, S., and M. Moher, Introduction to Analog & Digital 8 Communications, 2nd ed., Wiley, 2007. 4
5. Rectangular Pulse Amplitude Spectrum Spectrum of a Rectangular Pulse 2 1.5 Amplitude 1 0.5 0 -8 -6 -4 -2 0 2 4 6 8 Time 10 Amplitude Spectrum 5 0 -5 -1.5 -1 -0.5 0 0.5 1 1.5 frequency Haykin, S., and M. Moher, Introduction to Analog & Digital 9 Communications, 2nd ed., Wiley, 2007.Decaying Exponent Spectrum Decaying Exponential Pulse 1 0.8 Amplitude 0.6 0.4 0.2 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Amplitude Spectrum of Decaying Exponential Pulse 0.8 0.6 Magnitude 0.4 0.2 0 -3 -2 -1 0 1 2 3 frequency Phase Spectrum of Decaying Exponential Pulse 100 Phase in degrees 50 0 -50 -100 -3 -2 -1 0 1 2 3 frequency Haykin, S., and M. Moher, Introduction to Analog & Digital 10 Communications, 2nd ed., Wiley, 2007. 5
6. Rising Exponent Spectrum Rising Exponential Pulse 1 Amplitude 0.5 0 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Time Amplitude Spectrum of Rising Exponential Pulse 0.5 Magnitude 0 -3 -2 -1 0 1 2 3 frequency Phase Spectrum of Rising Exponential Pulse Phase in degrees 100 0 -100 -3 -2 -1 0 1 2 3 frequency Haykin, S., and M. Moher, Introduction to Analog & Digital 11 Communications, 2nd ed., Wiley, 2007.Properties of the FT Linearity c1 g1 (t ) + c2 g 2 (t ) ⇔ c1G1 ( f ) + c2G2 ( f ) Dilation 1 f g (at ) ⇔ G  a a Conjugation g * (t ) ⇔ G * (− f ) Duality If g (t ) ⇔ G ( f ), then G (t ) ⇔ g (− f ) Haykin, S., and M. Moher, Introduction to Analog & Digital 12 Communications, 2nd ed., Wiley, 2007. 6
7. Properties of the FT Time Shifting g (t − t0 ) ⇔ G ( f )e − j 2π ⋅ f ⋅t0 Frequency Shifting e j 2π ⋅ f c ⋅t g (t ) ⇔ G ( f − f c ) Differentiation dn n {g (t )} ⇔ ( j 2π ⋅ f )n ⋅ G( f ) dt Integration t 1 ∫ g (τ )dτ ⇔ −∞ j 2π ⋅ f G( f ) Haykin, S., and M. Moher, Introduction to Analog & Digital 13 Communications, 2nd ed., Wiley, 2007.Properties of the FT Area under g(t) ∞ −∞ ∫ g (t )dt = G (0 ) Area under G(f) ∞ g (0 ) = ∫ G ( f )df −∞ Modulation Theorem ∞ g1 (t )g 2 (t ) ⇔ ∫ G1 (λ )G2 ( f − λ )dλ −∞ Rayleigh’s Energy ∞ ∞ ∫ g (t ) ∫ G( f ) 2 2 Theorem dt = df −∞ −∞ Haykin, S., and M. Moher, Introduction to Analog & Digital 14 Communications, 2nd ed., Wiley, 2007. 7
8. Properties of the FT Haykin, S., and M. Moher, Introduction to Analog & Digital 15 Communications, 2nd ed., Wiley, 2007.FT Theorems Haykin, S., and M. Moher, Introduction to Analog & Digital 16 Communications, 2nd ed., Wiley, 2007. 8
9. FT Properties Examples Haykin, S., and M. Moher, Introduction to Analog & Digital 17 Communications, 2nd ed., Wiley, 2007.The Inverse Relationshipbetween Time and Frequency The properties of the FT show that the time-domain and frequency- domain description of a signal are inversely related to each other. − If the time-domain description of a signal is changed, the frequency- domain description of the signal is changed in an inverse manner, and vice versa. − A signal cannot be strictly limited in both time and frequency. Haykin, S., and M. Moher, Introduction to Analog & Digital 18 Communications, 2nd ed., Wiley, 2007. 9
10. BandwidthProvides a measure of the extend ofthe significant spectral content ofthe signal for positive frequencies.− A signal is low-pass if its significant spectral content is centered around the origin f = 0.− A signal is band-pass if its significant spectral content is centered around ±fc , where fc is a constant frequency. Haykin, S., and M. Moher, Introduction to Analog & Digital 19 Communications, 2nd ed., Wiley, 2007. BandwidthNull-to-null bandwidth− when the spectrum of the signal is symmetric with a main lobe bounded by well-defined nulls (i.e. frequencies at which the spectrum is zero), we may use the main lobe for defining the bandwidth of the signal.3-dB bandwidth− the separation (along the positive frequency axis) between the two frequencies at which the amplitude spectrum of the signal drops to 1/ 2 of the peak value. 20 10
11. Time-Bandwidth Product The product of the signal’s duration and its bandwidth is always a constant. (duration) X (bandwidth) = constant The time-bandwidth product is another manifestation of the inverse relationship that exists between the time-domain and frequency-domain descriptions of a signal. Haykin, S., and M. Moher, Introduction to Analog & Digital 21 Communications, 2nd ed., Wiley, 2007.Dirac Delta Function(Unit Impulse) The theory of the FT is applicable to only time functions that satisfy the Dirichlet conditions, but it would be helpful to extend the theory in two ways − To combine the theory of Fourier series and FT, so that the Fourier series may be treated as a special case of the FT. − To expand applicability of the FT to include power signals (periodic signals), signals that satisfy:  1 T  ∫ g (t ) dt  < ∞ 2 lim  T → ∞ 2T  −T  22 11
12. Dirac Delta Function This can be accomplished with the use of the Dirac Delta function. δ (0 ) = 0 , t ≠ 0 ∞ ∫ δ (t )dt −∞ =1 ∞ ∫ g (t )δ (t − t )dt −∞ 0 = g (t 0 ) (Sifting Property) ℑ { (t )} = 1 δ 23Applications of the DeltaFunction DC signal 1⇔ δ(f ) Complex Exponential e j 2π ⋅ f c ⋅t ⇔ δ ( f − f c ) Sinusoidal Functions 1 cos(2π ⋅ f c ⋅ t ) ⇔ [δ ( f − f c ) + δ ( f + f c )] 2 1 sin (2π ⋅ f c ⋅ t ) ⇔ [δ ( f − f c ) − δ ( f + f c )] 2j 24 12
13. Applications of the DeltaFunction Haykin, S., and M. Moher, Introduction to Analog & Digital 25 Communications, 2nd ed., Wiley, 2007. Dirac Delta Function Examples Haykin, S., and M. Moher, Introduction to Analog & Digital 26 Communications, 2nd ed., Wiley, 2007. 13
14. Fourier Transform of Periodic Signal Using the Fourier series, a periodic signal can be represented as a sum of complex exponential or into an infinite sum of sine and cosine terms. To denotes the period of the signal. fo denotes the fundamental frequency of the signal. 1 fo = To Haykin, S., and M. Moher, Introduction to Analog & Digital 27 Communications, 2nd ed., Wiley, 2007. FT of Periodic Signals ∞ ∞ x(t ) = ∑ g (t − mT ) ⇔ X ( f ) = f ∑ G(n ⋅ f )δ ( f − n ⋅ f ) 0 0 o 0 m = −∞ n = −∞ ∞ x(t ) = f 0 ∑ G (n ⋅ f )⋅ e 0 j 2π ⋅n⋅ f 0 ⋅t n = −∞ ∞x(t ) = f 0 ⋅ G (0 ) + 2 ⋅ f 0 ∑ G (n ⋅ f o ) ⋅ cos(2π ⋅ n ⋅ f 0 ⋅ t + ∠G (n ⋅ f o )) n =1 x(t) g(t) t T0 T0t1 t1+T0 28 14
15. Fourier Series: Example 1 Periodic Waveform 1.5 1 Amplitude 0.5 0 -0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 time (s) Haykin, S., and M. Moher, Introduction to Analog & Digital 29 Communications, 2nd ed., Wiley, 2007.Fourier Series Example 2 Periodic Waveform 1.5 1 0.5 Amplitude 0 -0.5 -1 -1.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 time (s) Haykin, S., and M. Moher, Introduction to Analog & Digital 30 Communications, 2nd ed., Wiley, 2007. 15
16. Power Spectral Density(PSD) Parserval’s Theorem – relates the energy associated with a time- domain function of finite energy to the Fourier transform of the function. To calculate the PSD, it’s necessary to assume a resistor of 1 (normalized). The PSD (energy) (in Watts / Hz) of a signal x(t) is Sx = X ( f ) 2 Haykin, S., and M. Moher, Introduction to Analog & Digital 31 Communications, 2nd ed., Wiley, 2007.Power Spectral Density(PSD) The average power (normalized) (in Watts) is ∞ Pave = ∫ S ( f ) df −∞ x Parseval’s Theorem(Periodic Signals) ∞ 2 Pave = ∑ X (n ⋅ f ) n = −∞ 0 Haykin, S., and M. Moher, Introduction to Analog & Digital 32 Communications, 2nd ed., Wiley, 2007. 16
17. Examples 1 and 2 (PSD) Example 1 PSD 0.4 0.3 Pave = 0.4833 W PSD Sx 0.2 0.1 0 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (Hz) Example 2 PSD 0.4 0.3 Pave=0.6464 W PSD Sx 0.2 0.1 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Frequency (Hz) Haykin, S., and M. Moher, Introduction to Analog & Digital 33 Communications, 2nd ed., Wiley, 2007. 17
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