2.1 Basic Concepts Basic operations on signals Classication of signals

Transcription

1 Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1 Basic Concepts Basic operations on signals 1. Time Shifting 2. Time reversal 3. Time scaling Classication of signals 1. Continuous time and discrete time signals 2. Real and complex signals: e.g. x(t) Ae j(2πf 0t+θ) 3. Deterministic and Random Signals (Chapter 5) 4. Periodic and nonperiodic signals 5. Causal and non causal signals 6. Even and odd signals

2 7. Hermitian symmetry 8. Energy Type Signals: E x < example: E x + 9. Power Type Signals: 0 < P x < Example 2.1.9, : x(t) 2 dt lim 1 P x lim T T +T/2 T/2 T +T/2 T/2 x(t) 2 dt (1) x(t) 2 dt (2) Some Important Signals 1. Sinusoidal 2. Complex exponential 2

3 3. Unit step 4. Rectangular Pulse 5. Triangular 6. Sinc 7. Impulse Classication of Systems An entity excited by an input signal and produces an output signal. y(t) S[x(t)] 1. Discrete Time and Continuous Time Systems 2. Linear and Nonlinear systems 3. Time-Invariant and Time-Varying Systems 4. Causal and Noncausal Systems: LTI Systems in the Time Domain Impulse response h(t) S[δ] Convolution: Relation of input and output of an LTI system 3

4 y(t) S[x(t)] S[ x(τ)δ(t τ)dτ] x(t) h(t) Example y(t) t x(τ)dτ 2.2 Fourier Series x(t): periodic signal with period T o if Dirichlet conditions are satised, then: where x(t) x n e j2π n To t (3) n 4

5 e j2π n To t : f o : x n 1 α+to x(t)e j2π n To t dt (4) T o (g: periodic signal example) α 2.3 Fourier Transform In communications systems input (information bearing signal) Linear System Output (distorted) To analyze the relations between input, output Fourier Transform is a useful tool Dirichlet Conditions: x(t): Fourier transform of a signal can be derived if the signal is, absolutely integrable: has nite number of maxima in a nite interval has nite number of minima in a nite interval X(f) F(x(t)) (5) x(t) F 1 (X(f)) (6) x(t) X(f) (7) In ELE 371 we used X(ω), where ω 2πf : angular frequency (rad/sec) X(ω) X(f) x(t) 1 2π x(t) x(t)e jωt dt (8) x(t)e j2πft dt (Analysis equation) (9) X(ω)e jωt dt (10) X(f)e j2πft dt (Synthesis equation let 2πf ω) (11) (12) Example 2.3.1: Dene, 1 t < 1/2 Π(t) 1/2 t 1/2 0 otherwise 5 (13)

6 Find F(Π(t)) and draw the spectrum, Example 2.3.2: x(t) δ(t)f? (Use sifting property) X(f) δ(f)f 1? Result: What does this mean? Fourier transform of real,even and odd signals: Using Euler's equation ( If x(t) is real 6

7 F(x(t)) X(f) x(t)e j2πft dt (14) x(t) cos(j2πf t)dt j x(t) sin(j2πf t)dt }{{}}{{} (15) (16) X( f) (17) (18) (19) We observe that: Re{X( f)} Re{X(f)}() (20) Im{X( f)} Im{X(f)}() (21) X( f) X(f) (22) X( f) X(f) (23) If x(t) is real and even (x( t) x(t)) Imaginary part is zero Real part is even x(t) real and even X(f) real and even x(t) real and odd X(f) purely imaginary and odd 7

8 Bandwidth of a signal: Basic Properties of Fourier Transform Suppose x 1 (t) X 1 (f) and x 1 (t) X 2 (f) 1. Linearity ax 1 (t) + bx 2 (t) ax 1 (f) + bx 2 (f) 2. Duality: x(t) X(f) X(t) x( f) X( t) x(f) 3. Time Shifting: x(t t 0 ) e j2πft 0 X(f) 4. Scaling: x(at) 1 a X( f a ) 8

9 5. Convolution: x(t) y(t) X(f)Y (f) 6. Modulation x(t)e j2πf 0t X(f f 0 ) 7. Parseval: x(t)y (t)dt X(f)Y (f)df 8. Rayleigh: Energy of the signal x(t) 2 dt X(f) 2 df 9. Autocorrelation: R X (τ) x(t)x (t τ)dt Using the convolution property 9

10 10. Conjugate: x ( t) X ( f) 11. Dierentiation: dx(t) dt j2πfx(f) 12. Table 2.1: Table of Fourier Transform pairs Fourier Transform of Periodic Signals Fourier transform of periodic signals involve impulse functions in the frequency domain (e.g. sin, cos, impulse train) Fourier series x(t) n x ne j2π n T t 0 n x nδ(f n T 0 ) Truncated signal x T0 (t) { x(t) T0 /2 < t T 0 /2 0 otherwise 10

11 x(t) n x T0 (t nt 0 )(shift&add) x(t) X(f) X(f) X(f) x n 1 X T0 ( n ) T 0 T 0 X(f) sum 1 n X T0 ( n )δ(f n ) T 0 T 0 T 0 Fourier series coecients can be found using the F.T Transmission over LTI systems : Input-output relationship: Convolution in time Y (f) X(f)H(f) in frequency domain. Frequency domain (multiplication) is usually much simpler than the time domain (convolution) Communications channel (e.g. multipath fading channel) Pulse shaping (e.g. raised cosine) Filter (e.g. low pass, high pass, bandpass) is a very common example of an LTI system Example : Input x(t) sinc(w 1 t) impulse response: h(t) sinc(w 2 t). Output? 11

12 Example : 1 Filter: H(f). Determine the lter type and its 3dB bandwidth. 1+( 10000) f Filter Design Filters separate signals from each other and from interference Impulse response and frequency response Analog (Passive and Active - ELE 202) and Digital Filters (ELE we are interested) Innite Impulse Response: Fewer coecients but phase distortion Finite Impulse Response: Linear phase characteristics- Preferred Filter parameters: 12

13 1. Passband edge frequency 2. Stopband edge frequency 3. Passband ripple (20 log 10 1+δ 1 1 δ 1 ) 4. Stopband attenuation (20 log 10 δ 2 1+δ 1 ) 5. Matlab "remez " algorithm hremez(m,f,m,w) 6. Mlength of the lter M 20 log 1 0( δ 1 δ 2 ) f f : transition bandwidth 8. f [0f p, f s 1] critical frequencies 9. m [] magnitude response 10. w [] weighting vector Magnitude characteristics of a physically realizable lter Example 2.4.1: Use remez function to design an FIR low pass lter that meets the following specications 1. Passband ripple 1dB 2. Stopband attenuation 40dB 3. Passband edge frequency

14 4. Stopband edge frequency 0.35 Solution: 2.5 Power and Energy We have dened the power and energy type signals before We will analyze the autocorrelation, power and energy spectra of these type of signals Energy-Type Signals Autocorrelation: R x (τ) x(τ) x ( τ) Set τ 0, E x Using the Fourier transform of autocorrelation or Rayleigh theorem E x Therefore, the energy spectral density of x(t) is equal to 14

15 G x This is also the Fourier transform of the autocorrelation function F{R x (τ)}. Signal passed through a system. Using Rayleigh theorem Y (f) X(f)H(f) R y (τ) F 1 [ Y (f) 2 ] R x (τ) R h (τ) G y Example 2.5.2:Autocorrelation function, energy spectral density and energy content of: x(t) e αt u(t), α > Power-Type Signals Time averaged autocorrelation function 15

16 Rx(τ) p 1 T/2 lim x(t)x (t τ)dt (24) T T T/2 P x (25) R p x(0) (26) S x (f) F{R p x(τ)}p owerspectraldensity (27) R p x(τ) R p x(0) When passed through a system, y(t) R y (τ) lim T S p x(f)e +j2πfτ df (28) S p x(f)df P x (29) x(τ)h(t τ)dτ T/2 T/2 y(t)y (t τ)dt (30) R x (τ) h(τ) h ( τ) S y (f) S x (f)h(f)h (f) 2.6 Properties of Hilbert Transform Unlike other transformations, Hilbert transform does not change the domain. Frequency components of Hilbert transform lags the original signal by 90 o. e j2πf 0t becomes e j2πf 0t π/2 je j2πf 0t and e j2πf 0t becomes e j2πf 0t π/2 je j2πf 0t. In frequency domain: multiplication by jsgn(f): F[ x(t)] 16

17 In time domain: x(t) Example 2.6.1: Find the Hilbert transform of x(t) 2sinc(2t) Important properties: 2.7 Lowpass and Bandpass signals Denition of a low pass signal: Denition of a band pass signal: Simplest band pass signal: In phase and quadrature components: x c (t) and x s (t) are base- For a general signal band signals. 17

18 Result: Spectrum of a bandpass signal 18