DSP-Lec 03-Discrete Time Systems

Chapter p Discrete-Time Systems Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn Content ™ Input/output I t/ t t relationship l ti hi off the th systems t ™ Linear time-invariant time invariant (LTI) systems ‰ convolution ™ FIR andd IIR filters fil ™ Causality C li and d stability bili off the h systems Ha H Kha Discrete-Time Systems Discrete-time signal ™ The discrete-time signal x(n) is obtained from sampling an analog signal x(t), (t) i.e., i e x(n)=x(nT) (n)= (nT) where here T is the sampling period period ™ There are some representations of the discrete-time signal x(n): x(n) ™ Graphical representation: ™ Function: ™ Table: T bl ⎧1 ⎪ x ( n) = ⎨ ⎪0 ⎩ n … x(n) … for n = 1,3 for n = ‐1 elsewhere l h 1 n ‐2 ‐1 … 0 0 … ™ Sequence: q x(n)=[… ( ) [ 0,, 0,, 1,, 4,, 1,, 0,, …]=[0, ] [ , 1,, 4,, 1]] Ha H Kha Discrete-Time Systems Some elementary of discrete-time signals ™ Unit sample sequence (unit impulse): ⎧1 δ ( n) = ⎨ ⎩0 for n = for n ≠ ™ Unit step signal ⎧1 u ( n) = ⎨ ⎩0 Ha H Kha f n≥0 for for n < Discrete-Time Systems Input/output rules ™ A discrete-time system is a processor that transform an input seq ence x(n) sequence (n) into an output o t sequence seq ence y(n) (n) Fig: Discrete-time system ™ Sample-by-sample Sample by sample processing: that is, and so on ™ Block processing: Ha H Kha Discrete-Time Systems Basic building blocks of DSP systems ™ Constant multiplier p ™ Delay D l y (n) = ax(n) x((n) y ( n) = x ( n − D ) x(n ( ) x2 ((n n) ™ Adder dde y (n) = x1 (n) + x2 (n) x1 (n) x2 ( n ) ™ Signal multiplier Ha H Kha x1 (n) y (n) = x1 (n) x2 (n) Discrete-Time Systems Example ™ Let x(n)={1, ( ) { , 3,, 2,, 5} } Find the output p and plot p the graph g p for the systems with input/out rules as follows: y( ) ( ) a)) y(n)=2x(n) b) y(n)=x(n-4) c) y(n)=x(n)+x(n-1) y(n)=x(n)+x(n 1) Ha H Kha Discrete-Time Systems Example ™ A weighted g average g system y y(n)=2x(n)+4x(n-1)+5x(n-2) y( ) ( ) ( ) ( ) Given the input signal x(n)=[x0,x1, x2, x4 ] p y(n) y( ) byy sample-sample p p p processingg method? a)) Find the output b) Find the output y(n) by block processing method c) Plot the block diagram to implement this system from basic building blocks ? Ha H Kha Discrete-Time Systems Linearity and time invariance ™ A linear system has the property that the output signal due to a linear combination of ttwo o inp inputt signals can be obtained b by forming the same linear combination of the individual outputs Fig: Testing linearity ™ If y(n)=a1y1(n)+a2y2(n) ∀ a1, a2 Ỉ linear system Otherwise, the system is nonlinear Ha H Kha Discrete-Time Systems Example ™ Test the linearity of the following discrete-time systems: a) y(n)=nx(n) b) y(n)=x(n2) c) y(n)=x2(n) d) y(n)=Ax(n)+B Ha H Kha 10 Discrete-Time Systems Linearity and time invariance ™ A time-invariant system is a system that its input-output characteristics not change with ith time time Fig: g Testingg time invariance ™ If yD(n)=y(n-D) ∀ DỈ time-invariant system Otherwise, the system is time-variant Ha H Kha 11 Discrete-Time Systems Example ™ Test the time-invariance of the following discrete-time systems: a) y(n)=x(n)-x(n-1) b) y(n)=nx(n) c) y(n)=x(-n) d) y(n)=x(2n) Ha H Kha 12 Discrete-Time Systems Impulse response ™ Linear time-invariant (LTI) systems are characterized uniquely by their impulse response sequence h(n), which is defined as the response of the systems to a unit impulse δ(n) Fig: Impulse response of an LTI system Fig: i Delayed D l d impulse i l responses off an LTI T system Ha H Kha 13 Discrete-Time Systems Convolution of LTI systems Fig: Response to linear combination of inputs ™ Convolution: y (n) = ∑ x(m)h(n − m) = x(n) ∗ h(n) (LTI form) m y (n) = ∑ h(m) x(n − m) = h(n) ∗ x(n) (direct form) m Ha H Kha 14 Discrete-Time Systems FIR and IIR filters ™ A finite impulse response (FIR) filter has impulse response h(n) that extend only over a finite time interval, interval say ≤n ≤ M M Fi FIR impulse Fig: i l response ™ M: filter order; Lh=M+1: the length g of impulse p response p ™ h={h0, h1, …, hM} is referred by various name such as filter coefficients, filter weights, or filter taps ™ FIR filtering equation: y (n) = h(n) ∗ x(n) = M ∑ h ( m) x ( n − m) m =0 Ha H Kha 15 Discrete-Time Systems Example ™ The third-order FIR filter has the impulse response h=[1, 2, 1, -1] a) Find the I/O equation, i.e., the relationship of the input x(n) and the output y(n) ? b) Given x=[1, 2, 3, 1], find the output y(n) ? Ha H Kha 16 Discrete-Time Systems FIR and IIR filters ™ A infinite impulse response (IIR) filter has impulse response h(n) of infinite duration, duration say ≤n ≤ ∞ ∞ Fi IIR impulse Fig: i l response ™ IIR filtering equation: y (n) = h( n) ∗ x(n) = ∞ ∑ h ( m) x ( n − m) m =0 ™ The I/O equation of IIR filters are expressed as the recursive difference equation Ha H Kha 17 Discrete-Time Systems Example ™ Determine the output of the LTI system which has the impulse r p n h(n)= response h(n)=anu(n), (n) |a|≤ | |≤ when h n th the inp inputt is i the th unit nit step t p signal i n l x(n)=u(n) ? ™ Remark: m n+ +1 r − r k r = ∑ 1− r k =m n ™ When n= ∞ and|r|≤ Ha H Kha ∞ m r k r = ∑ 1− r k =m 18 Discrete-Time Systems Example ™ Assume the IIR filter has a casual h(n) defined by for n = for n ≥ ⎧ h( n) = ⎨ n −1 ( ) ⎩ a)) Find Fi d the h I/O difference diff equation i ? b) Find the difference equation for h(n)? Ha H Kha 19 Discrete-Time Systems Causality and Stability Fig: Causal, anticausal, and mixed signals ™ LTI systems can also classified in terms of causality depending on whether h(n) is casual, anticausal or mixed ™ A system is stable (BIBO) if bounded inputs (|x(n)| ≤A) always generate bounded outputs (|y(n)| ≤B) ™ A LTI system is stable ⇔ ∞ ∑ | h( n) | < ∞ n = −∞ Ha H Kha 20 Discrete-Time Systems Example ™ Consider the causality and stability of the following systems: a) h(n)=(0.5)nu(n) b)) h(n)=-(0.5) ( ) ( )nu(-n-1) ( ) Ha H Kha 21 Discrete-Time Systems Homework ™ Problems: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 Ha H Kha 22 Discrete-Time Systems ... the th systems t ™ Linear time- invariant time invariant (LTI) systems ‰ convolution ™ FIR andd IIR filters fil ™ Causality C li and d stability bili off the h systems Ha H Kha Discrete -Time Systems. .. Discrete -Time Systems Example ™ Test the linearity of the following discrete -time systems: a) y(n)=nx(n) b) y(n)=x(n2) c) y(n)=x2(n) d) y(n)=Ax(n)+B Ha H Kha 10 Discrete -Time Systems Linearity and time. .. time- invariant system Otherwise, the system is time- variant Ha H Kha 11 Discrete -Time Systems Example ™ Test the time- invariance of the following discrete -time systems: a) y(n)=x(n)-x(n-1) b) y(n)=nx(n)