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Dsp U Lec04 Discrete Time Signals & Systems

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Discrete Time Signals & Systems

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Dsp U Lec04 Discrete Time Signals & Systems

1. 1. EC533: Digital Signal Processing Lecture 4 Discrete-Time Signals & Systems
2. 2. x(n) y(n) 4.1 – Discrete-Time Signals DSP By tradition, a discrete-time signal is represented as a sequence of numbers: x(n)=5,3,4,3,6,…. for n=0,1,2,… x(n) x(nT)=5,3,4,3,6,…. for n=0,1,2,… 6 5 4 xn=5,3,4,3,6,…., for n=0,1,2,… 3 3 T 2T 3T 4T nT where x(n) indicates the value of the signal at a discrete time n(nT), also, it may indicates the sequence itself. • In DSP, it is common to omit T as the sampling frequency is assumed to be unity.
3. 3. 4.1.1 – Important Discrete Signals As we are considering processing signals that are represented by sequences,  we shall introduce the following basic signals: a)Unit Impulse (Unit Sample) 1 0 n b)Unit Step 1 0 1 2 3 4 n
4. 4. 4.1.1 – Important Discrete Signals – cont. Delayed & Advanced Sequences 1 delayed For f(±n ±m); If n & m have the same sign, the sequence will be 0 2 n advanced by m samples (shifted left). If n & m have the opposite signs, the sequence will be delayed by m samples (shifted right). 1 advanced ‐1 0 n c) Unit Ramp r (n) ⎧n for n > 0⎫ r ( n) = ⎨ ⎬ ⎩0 otherwise⎭ 0 1 2 3 4 n
5. 5. 4.1.1 – Important Discrete Signals – cont. d) exponential signal: x(n) = a n for ∀n a = r.e jθ When |a|>1 The signal converges to 0 at ∞
6. 6. 4.1.1 – Important Discrete Signals – cont. Unit Impulse & Unit Step Relationship 1 0 1 2 3 4 n 1 1 2 3 4 n 1 1 0 1 2 3 4 n 0 n Hence, the unit impulse signal can be used as a basic building block for the construction  & representation of other signals. 3 2 1 0.5 ‐1 0 1 3 n
7. 7. 4.2 – Discrete-Time Systems • A discrete-time system is essentially a mathematical algorithm that takes an input sequence, x(n), & produces an output sequence y(n). e.g., digital controllers, digital spectrum analyzers, & digital filters. x(n) y(n) DT system • The discrete-time system is described by its impulse (unit sample) response h(n) if h(n) then h(n) Impulse Response Characteristics of discrete-time systems Linearity Shift Invariance (Time Invariance) Stability Causality
8. 8. 4.3 – Characteristics of discrete-time systems • A discrete system is linear if it satisfies the superposition principle, that is: If y1(n) is the o/p for the i/p x1(n) , y2(n) is the o/p for the i/p x2(n) then the o/p for the i/p α x1 (n) + β x2 (n) is α y1 (n) + β y2 (n) h(n) h(n) h(n)
9. 9. Linear System: Example 1 n Accumulator y ( n ) = ∑ x (l ) l = −∞ if x ( n) = α x1 (n) + β x2 (n) Then n y (n ) = ∑ α x1 (l ) + β x 2 (l ) l = −∞ n n = ∑ α x1 (l ) + ∑ β x 2 (l ) l = −∞ l = −∞ n n = α ∑ x1 (l ) + β ∑ x 2 ( l ) l = −∞ l = −∞ = α y1 ( n ) + β y 2 ( n )
10. 10. Linear System: Example 2 y [ n ] = ( x [ n ]) 2 if x(n) = α x1 (n) + β x2 (n) Then y [ n ] = ( α x 1 [ n ] + β x 2 [ n ]) 2 ≠ α y1[ n ] + β y 2 [ n ]
11. 11. 4.3.2 – Shift Invariance (Time Invariance) A system is shift invariant if any delay in the i/p produces a similar delay in the o/p. x1(n) y1(n) i.e if x1 ( n ) ⎯ ⎯→ y1 ( n ) h(n) then if x ( n ) = x1 ( n − n 0 ) y ( n ) = y1 ( n − n 0 ) x1(n-k) y1(n-k) h(n) i.e. process doesn’t depend on absolute value of n
12. 12. Shift Invariant system (Example 1) • Upsampler x(n) y (n) L Not shift‐invariant
13. 13. Shift Invariant system (Example 2) Scaling by the time index Hence, if then Not shift-invariant parameters depend on n
14. 14. Linear Shift Invariance (LSI) Systems • Systems which are both linear and shift invariant are easily  manipulated mathematically • If discrete index corresponds to time, called Linear Time  Invariant (LTI) • There is a wide and useful class of DSP systems as digital filters.
15. 15. Linear Time-Invariant (LTI) systems • LTI Systems can be fully characterized by the convolution sun •Since •Due to linearity and shift-invariance: •Then Convolution sum •The convolution describes how the I/p to a system interacts with the system to produce the O/p.
16. 16. Properties of Convolution Sum 1- Commutative: 2- Distributive: x(n) h1(n) y(n) h2(n) Parallel connection 3- Associative: x(n) y(n) x(n) y(n) h1(n) h2(n) Cascade connection
17. 17. LTI System Representation • LTI systems can be represented by 2 forms: 1. Convolution Sum: 2. Difference Equation: If b’s=0   FIR (Finite Impulse Response) system else        IIR (Infinite Impulse Response) system
18. 18. 4.3.3 –Stability • A system is stable if each Bounded Input produces a Bounded Output (BIBO). Bounded i/p : Bounded o/p : for Condition for Stability The system is stable if it is absolutely summable.
19. 19. 4.3.4 – Causality • A system is causal if there is no output when there is no input, The o/p of a causal system depends only on the present & past values of the i/p to the system & doesn’t predict future.
20. 20. Causality (Example) Moving Average depends on Causal ‘Centered’ Moving Average looks forward in time noncausal can be made causal by delaying