System Analysis System: Responds to signals and produces new signals Mostly commonly analyzed are artificial systems designed and built by human System analysis: application of System Analysis mathematical model to the design and analysis of systems Predict the output of a system for different kinds of signals using various 1 mathematical techniques 2 Model Examples A system can be classified as continuous- Continuous time to discrete signals time system or discrete-time system x(t) y[n] = x(nT ) H{ }: sampling s y(t) = H{x(t)} Continuous Time: x(t) y(t) Signal amplifier System H{ } Discrete: Time: x[n] y[n] x(t) y(t) = 10·x(t) y[n]= H{x[n]} H{ }: amplifier 3 4 Different kinds of System Additive System A system is additive if the following is true A system can be classified as If Memoryless: does not depend on past inputs y1(t) is the response of x1(t) Causal: does not depend on future inputs y2(t) is the response of x2(t) Additive x1(t) y1(t) x2(t) y2(t) Output for (x1+x2) = output for x1 + output for x2 H{ } H{ } Homogeneous Then Output for (ax1) = output of x1 scaled by a Linear: additive + homogeneous y1(t) + y2(t) is the response of x1(t) + x2(t) x1(t)+ x (t) y (t)+ y (t) Time invariant: behaviour does not change with 2 H{ } 1 2 time 5 6 Homogeneous System Linear System A system is homogeneous if the following A system is linear if it is both additive is true and homogeneous If x(t) y(t) Linearity is an extremely important y(t) is the response of x(t) T{ } property A huge amount of results on linear Then operations can be applied to study a ay(t) is the response of ax(t) system’s behaviors and structure a is a real scalar ax(t) ay(t) Nonlinear systems are often approximated H{ } by linear systems 7 8 Example: Linear Systems Example: Nonlinear Systems Amplify the Take the square input by 10 times of the input Assume Assume x1(t=1) = 0.5 y1(t=1) =5 x1(t=1) = 0.5 y1(t=1) =0.25 x2(t=1) = 0.3 y2(t=1) =3 x2(t=1) = 0.3 y2(t=1) =0.09 Then Then x3(t=1) = x1(t=1) + x2(t=1) = 0.8 x3(t=1) = x1(t=1) + x2(t=1) = 0.8 y3(t=1) =8 = y1(t=1) + y2(t=1) y3(t=1) =0.64 ≠ y1(t=1) + y2(t=1) = 0.34 9 10 Example: Nonlinear Systems Time invariant System Reason Time invariance: 2 If an input is delayed by t , then the output System: y(t) = { x(t) } 0 is also delayed by the same amount 2 {x1(t) + x2(t)} 2 2 = { x1 (t) } + { x2 (t) } + 2 x1(t) x2(t) 2 2 ≠ { x1 (t) } + { x2 (t) } 11 12 Example Linear time invariant systems Mute the input for 1 second in every cycle of 2 seconds Non-linear or time variant systems are Input Output 1 1 very difficult to analyze 0.8 0.6 0.4 0.5 0.2 0 0 -0.2 Many real-life systems can be approximated -0.4 -0.5 -0.6 -0.8 2 sec 1 sec -1 -10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 as linear and time invariant Time t (sec) 1 Input with 1 sec delay 1 Output with 1 sec delay Linear Time Invariant (LTI) Systems 0.8 0.6 0.4 0.5 0.2 0 0 -0.2 When a signal is fed into an LTI system, -0.4 -0.6 -0.5 -0.8 -1 -1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 the output can be described as the Output due to the input with 1 sec delay convolution integral between the input 1 0.5 signal and impulse response of the 0 -0.5 13 14 -1 13 system 0 1 2 3 4 5 6 7 8 9 10 Impulse Response Impulse response An impulse function (t) is defined as The impulse response of an LTI system is the response of the system when the input is an impulse function (t) We often use the impulse response to characterize an LTI system So the FT of (t) is 15 16 Convolution Integral Convolution Representation of LTI Continuous-Time Systems If the impulse response h(t) of an LTI system Unit impulse: (t) (t) 0, t 0 is known, the output of any input signal x(t) is the convolution integral of x(t) and h(t) (t)dt 1, for any 0 x(t) * h(t) Impulse Response: Input is (t), Output of the system is h[t] Input = (t) Output = h(t) (1) System 17 0 t 0 t 18 Convolution Representation of LTI Continuous-Time Systems Example For an arbitrary signal x(t), with x(t)=0 for t <0 x(t) = u(t) – 2u(t1) + u(t2) v(t) = u(t) – u(t1) x(t) x() (t )d 0 y(t)= x(t) v(t) = ? v(-) Input: (t), output: h(t) x() Input: (t -), output: h(t -) 1 1 y(t) x()h(t )d x(t)h(t) 012 -2 -1 0 0 -1 Convolution representation of the system The system is completely determined by h(t) 19 20 1 t For t [1,2], x(t) v(t) 1d 1d 2t 3 Example x() v(t) t-1 x1()v(t) 1 1 1 t = t For t [0,1], x(t) v(t) 1d t 012 01t 2 012 0 -1 t1 t1 x() v(t) x()v(t) 2 1 1 1 For t [2,3], x(t) v(t) 1d t 3 = t-1 x() v(t) x()v(t) 012 -1 0 1 -1 0 1 1 1 1 = -1 t v(t)= v(( t)) t t1 012 01 2 t 01 2 -1 t1 21 22 Convolution in Frequency x(t) v(t) Domain -2t+3 t 1 Convolution in time domain becomes multiplication in frequency domain 012 4t yt() xt ht () Y () X () H () -1 t-3 Proof y(t) t u(t) u(t 1) (2t 3) u(t 1) u(t 2) jt jt Yytedthtxdedt() (t 3)u(t 2) u(t 3) jt Yxhtdedttt , y(t) tu(t) (3t 3)u(t 1) jjt (3t 6)u(t 2) (t 3)u(t 3) YxedhtedtXH () 23 24 Convolution in Frequency Determine the LTI System Domain Output using Fourier Transform Use multiplication instead of convolution 25 26 Example: Example: x(t) 1 x(t) 1 a H() a j -0.5 0 0.5 Table -0.5 0 0.5 jt sin0.5 X x(t)e dt p1t sinc 2 0.5 x t X X Y Y 27 28 Example: Example: x(t) x(t) 1 1 a H() a j -0.5 0 0.5 Table -0.5 0 0.5 0.5 jt jt sin0.5 X x(t)e dt e dt p1t sinc 2 0.5 0.5 xt p1 t 1 jt 0.5 1 j0.5 j0.5 2sin(0.5) X e e e 2sin(0.5) j 0.5 j X 2sin(0.5) a 2sin(0.5) a Y Y a j a j 29 30 Example: Example: 2sin(0.5) a 2sin(0.5) 1 Y a 1, Y 180° a j 1 j Input 31 32 Example: 2sin(0.5) a 2sin(0.5) 10 a t Y a 10, Y ae , t 0 a j 10 j h(t) 0, t0 a = 1 a = 10 33 34 Applications of LTI system Lowpass Filters Frequency filtering Both anti-aliasing filter and post-filter are lowpass filters The process of rejecting sinusoids having particular frequencies, or for a range of Only let go low frequency information and frequencies block high frequency information A system having this characteristics is called a frequency filter Pre- A/D Digital Signal D/A Post- filter Processor filter 35 36 Lowpass Filters Practical Lowpass Filters Convolution in time domain is Ideal lowpass filter: sharp transition multiplication in frequency domain from 1 to 0 at frequency ||=B Practical lowpass filters have finite H transition regions Hmax Hmax 2 0 37 Bandwidth 38 Highpass filters Only let go high frequency information and block low frequency information 39 40 41 42 Noise removal Common use of filters: remove noise Noise bandwidth is much greater than the signal bandwidth 43 44 Summary References M.J. Roberts, Fundamentals of System properties Signals & Systems, McGraw-Hill, Linear time-invariant system 2008. System output = Chapters 4, 6 and 12 convolution of the input signal and the impulse response in time domain James H. McClellan, Ronald W. Multiplication of the FT of the input Schafer and Mark A. Yoder, Signal signal and the transfer function in Processing First, Prentice-Hall, 2003. frequency domain Chapters 9, 10 Filter response 45 46.