#### Transcript Continuous-Time Signal Analysis: The Fourier Transform

ENGR 4323/5323 Digital and Analog Communication Ch 3 Analysis and Transmission of Signals Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr Chapter Outline • • • • • • • • Aperiodic Signal Representation by Fourier Integral Fourier Transform of Useful Functions Properties of Fourier Transform Signal Transmission Through LTIC Systems Ideal and Practical Filters Signal Distortion over a Communication Channel Signal Energy and Energy Spectral Density Signal Power and Power Spectral Density The Fourier Transform Spectrum Fourier transform (FT) allows us to represent aperiodic (not periodic) signal in term of its frequency ω. x(t) The Fourier transform integrals: X ( ) j t x ( t ) e dt X ( ) X ( ) e X ( ) |X(ω)| ω The Fourier Transform Spectrum X ( ) X () X () e The Phase Spectrum The Amplitude (Magnitude) Spectrum The amplitude spectrum is an even function and the phase is an odd function. The Inverse Fourier transform: 1 x(t ) 2 jt X ( ) e d Useful Functions Unit Gate Function t rect 0 0.5 1 | t | / 2 | t | / 2 | t | / 2 1 -/2 /2 t /2 t Unit Triangle Function t 0 1 2 t / | t | / 2 | t | / 2 1 -/2 Useful Functions Interpolation Function sin t sinc(t ) t sinc(t ) 0 for t k sinc(t ) 1 for t 0 sinc(t) t Example Find the FT, the magnitude, and the phase spectrum of x(t) = rect(t/). Answer X ( ) /2 rect(t / )e jt dt sinc( / 2) /2 What is the bandwidth of the above pulse? The spectrum of a pulse extend from 0 to . However, much of the spectrum is concentrated within the first lobe (=0 to 2/) Examples Find the FT of the unit impulse (t). Answer X ( ) (t )e jt dt 1 Find the inverse FT of (). Answer 1 x(t ) 2 1 ( )e d 2 jt so thespectrumof a constantis an impulse 1 2 ( ) Examples Find the FT of the unit impulse train T (t ) 0 Answer 1 T0 (t ) T0 n jn0t e n 2 X ( ) T0 n ( n ) n 0 Properties of the Fourier Transform • Linearity: • Let then xt X and yt Y xt yt X Y • Time Scaling: • Let then xt X 1 xat X a a Compression in the time domain results in expansion in the frequency domain Internet channel A can transmit 100k pulse/sec and channel B can transmit 200k pulse/sec. Which channel does require higher bandwidth? Properties of the Fourier Transform • Time Reversal: • Let then xt X x(t ) X ( ) • Left or Right Shift in Time: • Let xt X then xt t0 X e jt0 Time shift effects the phase and not the magnitude. Properties of the Fourier Transform • Multiplication by a Complex Exponential (Freq. Shift Property): • Let then xt X x(t )e j0t X ( 0 ) • Multiplication by a Sinusoid (Amplitude Modulation): Let then xt X 1 xt cos 0t X 0 X 0 2 cos0t is the carrier, x(t) is the modulating signal (message), x(t) cos0t is the modulated signal. Example: Amplitude Modulation x(t) Example: Find the FT for the signal A x(t ) rect(t / 4) cos10t -2 2 Amplitude Modulation Modulation AM (t ) m(t ) cosct Demodulation AM (t ) cos 2 c t 0.5m(t )[1 cos 2c t ] Then lowpass filtering Properties of the Fourier Transform • Differentiation in the Frequency Domain: • Let then xt X n d n n t x(t ) ( j ) X ( ) n d • Differentiation in the Time Domain: Let then xt X n d n x(t ) ( j ) X ( ) n dt Example: Use the time-differentiation property to find the Fourier Transform of the triangle pulse x(t) = (t/) Properties of the Fourier Transform • Integration in the Time Domain: Let xt X t Then 1 x( )d j X ( ) X (0) ( ) • Convolution and Multiplication in the Time Domain: Let xt X yt Y Then x(t ) y(t ) X ( )Y ( ) 1 x1 (t ) x2 (t ) X 1 ( ) X 2 ( ) 2 Frequency convolution Example Find the system response to the input x(t) = e-at u(t) if the system impulse response is h(t) = e-bt u(t). Properties of the Fourier Transform • Parseval’s Theorem: since x(t) is non-periodic and has FT X(), then it is an energy signals: 1 E xt dt 2 2 X 2 d Real signal has even spectrum X()= X(-), E 1 X d 2 0 Example Find the energy of signal x(t) = e-at u(t) Determine the frequency so that the energy contributed by the spectrum components of all frequencies below is 95% of the signal energy EX. Answer: = 12.7a rad/sec 1 1 𝑥 −1 dx = 𝑡𝑎𝑛 𝑎2 + 𝑥 2 𝑎 𝑎 Properties of the Fourier Transform • Duality ( Similarity) : • Let then xt X X (t ) 2 x( ) Signal Transmission Through a Linear System Distortionless Transmission (System) Slope is constant for distortionless system Example 3.16 A transmission medium is modeled by a simple RC low-pass filter shown below. If g(t) and y(t) are the input and the output, respectively to the circuit, determine the transfer function H(f), θh(f), and td(f). For distortionless transmission through this filter, what is the requirement on the bandwidth of g(t) if amplitude response variation within 2% and time delay variation within 5% are tolerable? What is the transmission delay? Find the output y(t). 𝑑 𝑎 −1 𝑡𝑎𝑛 𝑎𝑥 = 𝑑𝑥 1 + 𝑎2 𝑥 2 Ideal Versus Practical Filters wR(t) t H(ω)*WR(t) h(t)wR(t) Ideal Versus Practical Filters Signal Distortion Over a Communication Channel 1. Linear Distortion 2. Channel Nonlinearities 3. Multipath Effects 4. Fading Channels - Channel fading vary with time. To overcome this distortion is to use automatic gain control (AGC) Linear Distortion Channel causes magnitude distortion, phase distortion, or both. Example: A channel is modeled by a low-pass filter with transfer function H(f) give by −𝑗2𝜋𝑓𝑡𝑑 1 + 𝑘𝑐𝑜𝑠2𝜋𝑓𝑇 𝑒 𝐻(𝑓) = 0 𝑓 <𝐵 𝑓 >𝐵 A pulse g(t) band-limited to B Hz is applied at the input of this filter. Find the output y(t). Nonlinear Distortion y(t) = f(g(t)) f(g) can be expanded by Maclaurin series y 𝑡 = 𝑎0 + 𝑎1 𝑔 𝑡 + 𝑎2 𝑔2 𝑡 + ⋯ + 𝑎𝑘 𝑔𝑘 𝑡 If the bandwidth of g(t) is B Hz then the bandwidth of y(t) is kB Hz. Example: The input x(t) and the output y(t) of a certain nonlinear channel are related as y(t) = x(t) + 0.000158 x2(t) Find the output signal y(t) and its spectrum Y(f) if the input signal is x(t) = 2000 sinc(2000t). Verify that the bandwidth of the output signal is twice that of the input signal. This is the result of signal squaring. Can the signal x(t) be recovered (without distortion) from the output y(t)? Continue Example Distortion Caused by Multipath Effects 𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 + α𝑒 −𝑗2𝜋𝑓(𝑡𝑑 +∆𝑡) 𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 (1 + α𝑒 −𝑗2𝜋𝑓∆𝑡 ) 𝐻 𝑓 = 𝑒 −𝑗2𝜋𝑓𝑡𝑑 (1 + α 𝑐𝑜𝑠2𝜋𝑓∆𝑡 − 𝑗α 𝑠𝑖𝑛2𝜋𝑓∆𝑡) 𝐻 𝑓 = 1+ α2 + 2α 𝑐𝑜𝑠2𝜋𝑓∆𝑡 𝑒𝑥𝑝 −𝑗 2𝜋𝑓𝑡𝑑 + 𝑡𝑎𝑛−1 α 𝑠𝑖𝑛2𝜋𝑓∆𝑡 1 + α 𝑐𝑜𝑠2𝜋𝑓∆𝑡 Common distortion in this type of channel is frequency selective fading Energy and Energy Spectral Density ∞ 𝐸𝑔 = 𝑔 𝑡 𝑔∗ 𝑡 𝑑𝑡 Energy in the time domain 𝐺(𝑓) 2 𝑑𝑓 Energy in the frequency domain −∞ ∞ 𝐸𝑔 = −∞ Energy spectral density (ESD), Ψ𝑔 (𝑓), is the energy per unit bandwidth (in hertz) of the spectral components of g(t) centered at frequency f. Ψ𝑔 (𝑓) = 𝐺(𝑓) 2 The ESD of the system’s output in term of the input ESD is Ψ𝑥 (𝑓) 𝐻(𝑓) Ψ𝑦 (𝑓) = 𝐻(𝑓) 2 Ψ𝑥 (𝑓) Essential Bandwidth of a Signal Estimate the essential bandwidth of a rectangular pulse g(t) = (t/T), where the essential bandwidth must contain at least 90% of the pulse energy. ∞ 𝐸𝑔 = −∞ 𝐵 𝐸𝐵 = 𝑔2 𝑡 𝑑𝑡 = 𝑇/2 𝑑𝑡 = 𝑇 −𝑇/2 𝑇 2 𝑠𝑖𝑛𝑐 2 𝜋𝑓𝑇 𝑑𝑓 = 0.9𝑇 −𝐵 B = 1/T Hz Energy of Modulated Signals The modulated signal appears more energetic than the signal g(t) but its energy is half of the energy of the signal g(t). Why? 𝜑 𝑡 = 𝑔 𝑡 𝑐𝑜𝑠2𝜋𝑓0 𝑡 1 Φ(𝑓) = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 ) 2 1 Ψ𝜑 (𝑓) = 𝐺 𝑓 + 𝑓0 + 𝐺(𝑓 − 𝑓0 ) 2 4 If f0 > 2B then 1 1 Ψ𝜑 𝑓 = Ψ𝑔 𝑓 + 𝑓0 + Ψ𝑔 (𝑓 + 𝑓0 ) 4 4 1 𝐸𝜑 = 𝐸𝑔 2 Time Autocorrelation Function and Energy Spectral Density The autocorrelation 𝜓𝑔 (𝜏) of a signal g(t) and its ESD Ψ𝑔 (𝑓) form a Fourier transform pair, that is 𝜓𝑔 𝜏 𝐹𝑇 𝑎𝑛𝑑 𝐼𝐹𝑇 Ψ𝑔 (𝑓) Example: Find the time autocorrelation function of the signal g(t) = e-atu(t), and from it determine the ESD of g(t). Signal Power and Power Spectral Density Power Pg of the signal g(t) 1 𝑃𝑔 = lim 𝑇→∞ 𝑇 𝑇/2 𝑔 𝑡 𝑔∗ 𝑡 𝑑𝑡 −𝑇/2 𝐸𝑔𝑇 𝑃𝑔 = lim 𝑇→∞ 𝑇 Power spectral density Sg(f) of the signal g(t) 𝐺𝑇 (𝑓) 𝑆𝑔 (𝑓) = lim 𝑇→∞ 𝑇 ∞ 2 𝑃𝑔 = ∞ 𝑆𝑔 𝑓 𝑑𝑓 = 2 −∞ 𝑆𝑥 (𝑓) 𝐻(𝑓) 𝑆𝑦 (𝑓) = 𝐻(𝑓) 2 𝑆𝑥 (𝑓) 𝑆𝑔 𝑓 𝑑𝑓 0 Time Autocorrelation Function of Power Signals Time autocorrelation Rg( ) of a power signal g(t) 1 ℛ𝑔 (𝜏) = lim 𝑇→∞ 𝑇 1 ℛ𝑔 (𝜏) = lim 𝑇→∞ 𝑇 𝑇/2 𝑔 𝑡 𝑔(𝑡 − 𝜏)𝑑𝑡 −𝑇/2 ∞ 𝑔𝑇 (𝑡)𝑔𝑇 (𝑡 + 𝜏)𝑑𝑡 −∞ 𝜓𝑔𝑇 (𝜏) ℛ𝑔 (𝜏) = lim 𝑇→∞ 𝑇 ℛ𝑔 (𝜏) 𝐹𝑇 𝑎𝑛𝑑 𝐼𝐹𝑇 𝑆𝑔 (𝑓) Autocorrelation a Powerful Tool If the energy or power spectral density can be found by the Fourier transform of the signal g(t) then why do we need to find the time autocorrelation? Ans: In communication field and in general the signal g(t) is not deterministic and it is probabilistic function. Example A random binary pulse train g(t). The pulse width is Tb/2, and one binary digit is transmitted every Tb seconds. A binary 1 is transmitted by positive pulse, and a binary 0 is transmitted by negative pulse. The two symbols are equally likely and occur randomly. Determine the PSD and the essential bandwidth of this signal. 1 1 0 1 0 0 1 0 Challenge: g(t) is not deterministic and can not be expressed mathematically to find the Fourier transform and PSD. g(t) is random signal. 1 ℛ𝑔 (𝜏) = lim 𝑇→∞ 𝑇 𝑇/2 𝑔 𝑡 𝑔(𝑡 − 𝜏)𝑑𝑡 −𝑇/2 Tb g(t) t 𝜏 0 For 0 < 𝜏 < 𝑇𝑏 /2 1 𝑇𝑏 1 𝜏 ℛ𝑔 𝜏 = lim −𝜏 𝑁 = − 𝑁→∞ 𝑁𝑇𝑏 2 2 𝑇𝑏 For 𝜏 > 𝑇𝑏 /2 ℛ𝑔 𝜏 = 0 g(t-τ) Homework Problem 𝑇𝑏 /2 < 𝜏 < 𝑇𝑏 (𝑘 + 1)𝑇𝑏 𝜏 𝑘𝑇𝑏 + 𝜏 + 𝑇𝑏 /2 𝑘𝑇𝑏 𝑘𝑇𝑏 + 𝜏 1 𝑇𝑏 𝑁 ℛ𝑔 𝜏 = lim 𝑘𝑇𝑏 + 𝜏 + − (𝑘 + 1)𝑇𝑏 𝑁→∞ 𝑁𝑇𝑏 2 4 1 𝜏 1 ℛ𝑔 𝜏 = − 4 𝑇𝑏 2 ℛ𝑔 𝜏 1/8 −𝑇𝑏 −𝑇𝑏 /2 𝑇𝑏 /2 𝑇𝑏 Discrete Fourier Transform (DFT, FFT) 𝑇0 𝑁0 = 𝑇𝑠 2𝜋 Ω0 = 𝑁0 𝑁0 −1 𝑔𝑘 𝑒 −𝑗𝑞Ω0 𝑘 𝐺𝑞 = 𝑘=0