![Linear and non linear system
A System is said to be linear if it follows super position
principle is called as linear other wise non-linear.
T[ax1(t)+bx2(t)]=aT[x1(t)]+bT[x2(t)]
Similarly for a discrete-time linear system,
T[ax1(n)+bx2(n)]=aT[x1(n)]+bT[x2(n)]](https://image.slidesharecdn.com/digitalsignalprocessing-171227110317/85/Digital-signal-processing-9-320.jpg)
![CONVOLUTION
Convolution is the process by which an input interacts
with an LTI system to produce an output
Convolution between of an input signal x[n] with a
system having impulse response h[n] is given as,
where * denotes the convolution
x [ n ] * h [ n ] = x [ k ] h [ n - k ]](https://image.slidesharecdn.com/digitalsignalprocessing-171227110317/85/Digital-signal-processing-12-320.jpg)
![EXAMPLE
Convolution
We can write x[n] (a periodic function) as an infinite
sum of the function xo[n] (a non-periodic function)
shifted N units at a time](https://image.slidesharecdn.com/digitalsignalprocessing-171227110317/85/Digital-signal-processing-13-320.jpg)
![Properties of convolution
Commutative…
x1[n]* x2[n] = x2[n]* x1[n]
Associative…
{x1[n]* x2 [n]}* x3[n] = x1[n]*{x2 [n]* x3[n]}
Distributive…
{x1[n] *x2[n]}* x3[n] = x1[n]x3[n] * x2[n]x3[n]](https://image.slidesharecdn.com/digitalsignalprocessing-171227110317/85/Digital-signal-processing-14-320.jpg)
![EXAMPLE OF DFT
Find X[k]
We know k=1,.., 7; N=8](https://image.slidesharecdn.com/digitalsignalprocessing-171227110317/85/Digital-signal-processing-20-320.jpg)
Digital signal processing
- 1. Digital Signal Processing . Mrs.R.Chitra, Assistant Professor(SS), Department of ECE, Faculty of Engineering, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore
- 2. SYSTEM DEFINITION: Combination of set of elements. It will process or manipulate over input signal to produce output signal
- 3. Classification of systems Continuous time and discrete time signal Lumped-parameter and distributed -parameter systems Static and dynamic systems Causal and non-causal systems Linear and non-linear systems Time variant and time -invariant systems Stable and unstable system
- 4. Continuous time system A Continuous time system is one which operates on a continuous time input signal and produces a continuous time output signal. y(t)= Tx(t)
- 5. Discrete time system A discrete time system is the one which operates on a discrete- time input signal and produces a discrete -time output signal. y(n)=T(X(n)
- 6. Lumped and distributed parameter In lumped parameter system each component is lumped at one point in space. In distributed parameter systems the signals are functions of space as well as time.
- 7. Static and dynamic system A system is called static or memory less if its output at any instant depends on the input at that instant. A system output is depends up on the past and future values of input.
- 8. Causal and non causal system A causal system is one for which the output at any time t depends on the present and past inputs. A non-causal system output depends on the future values of inputs.
- 9. Linear and non linear system A System is said to be linear if it follows super position principle is called as linear other wise non-linear. T[ax1(t)+bx2(t)]=aT[x1(t)]+bT[x2(t)] Similarly for a discrete-time linear system, T[ax1(n)+bx2(n)]=aT[x1(n)]+bT[x2(n)]
- 10. Stable and unstable system If the system has to be stable if and only if every bounded input produces bounded output. If system has to be unstable if bounded input produces unbounded output
- 11. Time variant and invariant system Time variant system have same input and output, y(n,k)=y(n-k). Time invariant system has no same input and output.
- 12. CONVOLUTION Convolution is the process by which an input interacts with an LTI system to produce an output Convolution between of an input signal x[n] with a system having impulse response h[n] is given as, where * denotes the convolution x [ n ] * h [ n ] = x [ k ] h [ n - k ]
- 13. EXAMPLE Convolution We can write x[n] (a periodic function) as an infinite sum of the function xo[n] (a non-periodic function) shifted N units at a time
- 14. Properties of convolution Commutative… x1[n]* x2[n] = x2[n]* x1[n] Associative… {x1[n]* x2 [n]}* x3[n] = x1[n]*{x2 [n]* x3[n]} Distributive… {x1[n] *x2[n]}* x3[n] = x1[n]x3[n] * x2[n]x3[n]
- 15. Linear convolution )(*)()()()( 2113 nxnxmnxmxnx m )()( 11 j eXnx FT )()( 22 j eXnx FT )()()()(*)()( 213213 jjj eXeXeXnxnxnx FT
- 16. Circular convolution Circular convolution of of two finite length sequences 16 1 0 213 N m Nmnxmxnx 1 0 123 N m Nmnxmxnx
- 17. Circular convolution )()())(()()( 21 1 0 13 nxnxmnxmxnx N m N )()( 11 kXnx DFT )()( 22 kXnx DFT )()()()()()( 213213 kXkXkXnxnxnx DFT both of length N
- 18. EXAMPLE )()( 01 nnnx )(2 nx )(*)( 21 nxnx 0 0 N 0 n0 N )()( 21 nxnx n0=2, N=7 0
- 19. DISCRETE FOURIER TRANSFORM It turns out that DFT can be defined as Note that in this case the points are spaced 2pi/N; thus the resolution of the samples of the frequency spectrum is 2pi/N. We can think of DFT as one period of discrete Fourier series
- 20. EXAMPLE OF DFT Find X[k] We know k=1,.., 7; N=8
- 21. PROPERTIES OF DFT Linearity Duality Circular shift of a sequence Copyright (C) 2005 Güner Arslan kbXkaXnbxnax kXnx kXnx 21 DFT 21 2 DFT 2 1 DFT 1 mN/k2jDFT N DFT ekX1-Nn0mnx kXnx N DFT DFT kNxnX kXnx
- 22. INVERSE OF DFT We can obtain the inverse of DFT Note that