DSP Discrete-time linear time-invariant (LTI) systems

1. EMT 488/3 Digital Signal Processing DSPDiscrete-time linear time-invariant (LTI) systems

2. LTI system • A linear, time-invariant (LTI) system must satisfy both the linearity and time-invariant property, as discussed in the beginning of the course. • The behavior of a discrete-time LTI system can be analyzed via • Convolution • Difference equation with constant coefficients • In this topic, we will focus on analysis of LTI systems in time domain.

3. Convolution • A discrete-time input signal with a sequence x(n) can be represented as a sum of weighted impulse sequences. • As an example, x(n) = {2, 4, 0, 3}, starting at n = -1, can be represented as x(n) = 2δ(n+1) + 4δ(n) + 3δ(n-2). • In general, an input sequence x(n) can be represented as • The output (response) of the system y(n) is given by

4. Convolution cont … • For linear systems, applying the linearity property yields • Then, applying the scaling property yields • where Γ[δ(n-k)] is the impulse response of the LTI system.

5. Convolution cont … • Assume that h(n, k) = Γ[x(n – k)]. • For time-invariant systems, Γ[x(n – k)] = h(n – k). • Therefore, for a discrete-time LTI system, • The equation shows that for a discrete-time LTI system, the response y(n) can be completely characterized if the single function h(n), which is the response to the input of a unit impulseδ(n), is known.

6. Properties of convolution • Given that • Assuming n – k = m and substituting n – k = m into the equation above yields • This proof suggests that convolution satisfies the commutative law i.e.

7. Properties of convolution cont .. • The commutative property of convolution suggest that either sequences h(n) or x(n) can be folded and shifted, and the other sequence is unchanged while computing y(n). In either way, the final response y(n) is identical. • Besides that, convolution also obeys the • Distributive law • Associative law

8. Properties of convolution cont .. • From the associative law, • This is important because it means systems in series can be reordered. This will be significant for direct form II structure implementation of LTI systems, which will be described in the following.

9. Convolution cont … • How to compute the output y(n0) at some instant say n = n0 using convolution? From the convolution formula, the response at n = n0 is given by • The steps required are according to the sequence given below: • Folding. Fold h(k) about k = 0 to obtain h(-k). • Shifting. Shift h(-k) by n0 to the right (left) if n0 is positive (negative), to obtain h(n0 - k). • Multiplication. Multiply x(k) by h(n0 - k) to obtain x(k)h(n0 - k). • Summation. Sum all the values of the product sequence x(k)h(n0 - k) to obtain y(n) at time n = n0.

10. Tutorial • The impulse response of a linear time-invariant system is Determine the response of the system if the input signal • Determine the output y(n) of a relaxed, linear time invariant system with impulse response when the input is a unit step sequence, that is

11. Finite-duration and infinite-duration impulse response • LTI systems can be further divided into two types i.e. • Finite impulse response (FIR) • Infinite impulse response (IIR) • In a FIR or finite-duration impulse response system, h(n) has a finite number of non-zero values over a finite time duration, and the rest of the values are all zero outside the finite time duration. Example: h(n) = {0, 3, 4, 5, 2, 1}. • In an IIR or infinite-duration impulse response system, h(n) has an infinite number of non-zero values over an infinite time duration. Example: h(n) = 0.2nu(n), which extends to infinite n.

12. Implementation of a FIR system • Let’s consider a FIR system with M+1 samples and convolution sum given by Graphical representation Require: M + 1 multiplications; M additions; M memory elements. The delay z-1 is a memory element.

13. Implementation of FIR system cont … • As shown in the example, note that to compute y(n), we only perform M+1 multiplications, M additions, and require M delay elements. In other words, we only need to perform convolution over the finite length (M+1) of h(n). Thus, a FIR system can be readily implemented. • Note that as M grows to infinity, the impulse response h(n) becomes infinite and we have an IIR system. • Unfortunately, IIR systems cannot be practically implemented by convolution.

14. Implementation of an IIR system • One method to implemented an IIR system is to use perform the calculation recursively. As an example, let’s consider a system with an input signal x(n) and impulse response h(n) = 0.5nu(n).

15. Implementation of an IIR system • In the previous example, note that the output y(n) is feed back into the input x(n). • The feedback loop contains a delay element, which is crucial for the realizability of the system. The absence of this delay will force the system to compute y(n) in terms of y(n), which is not possible for discrete-time systems. • The implementation of such IIR systems requires only 1 multiplication, 1 addition and 1 memory element, making it practically realizable.

16. Recursive and nonrecursive discrete-time systems • A recursive system is a system whose output y(n) at time n depends on past outputs such as y(n-1), y(n-2), and also inputs. • A nonrecursivesystem is a system whose output y(n) at time n only depends on the inputs. • The basic difference between recursive and nonrecursive systems is the feedback loop in recursive system, which feed back the output of the system into the input. • A FIR system described by convolution is nonrecursive. • Both FIR and IIR systems can be implemented recursively with difference equation, which will be described later.

17. Linear constant-coefficient difference equation • In general, discrete-time LTI systems can be described by linear constant-coefficient difference equations, which is given by an example of a general expression • The integer N is called the order of the difference equation or the order of the system. • The 1st term is the response due to the previous outputs of the system and the 2nd term is due to the input signal.

18. Direct form I LTI system implementation

19. Direct form II LTI system implementation ***associativity and commutative law

20. Direct form II LTI system implementation

21. Linear constant-coefficients difference eqn • To reiterate, LTI systems described by linear constant-coefficient difference equations is given by • The total response y(n) of the system can be expressed as • The response yzi(n) of the system with zero inputs i.e. x(n) = 0 for all n is called zero-input response or natural response, which depends on the nature of the system and the initial conditions. • When the system is initially relaxed at n = 0, it starts with zero initial condition i.e. all previous outputs are 0.

22. Linear constant-coefficients difference eqn • Under zero initial condition, the response yzs(n) is called zero state response or forced response, which depends on the nature of the system, zero initial conditions and input signal. • Next, our goal is to determine the response y(n) of the system given a specific input x(n) and a set of initial conditions. • The direct solution method assumes that the total response y(n) is given by where yh(n) is known as the homogeneous solution and yp(n) is known as the particular solution.

23. Linear constant-coefficients difference eqn • The solution yh(n) is obtained for zero inputs i.e. x(n) and all the past inputs are set to 0 for all n. • By setting x(n) and past inputs to 0 for all n in the linear constant-coefficients difference equation and assuming the solution yh(n) = n, yh(n) is given by • which can be written as • The term in the parentheses is called characteristic polynomial.

24. Linear constant-coefficients difference eqn • The solution yp(n), which depends on the specific input signal and is assumed to have a form dependent on the form of the input signal, must satisfy • Note: • The zero input response, yzi(n) is obtained from solution yh(n), which depends on the initial conditionsgiven. • The zero state response, yzs(n) is obtained from solution yh(n) + yp(n), which depends on the input signal given and initial conditions set to zero.

25. General form of the particular solution for several types of input signals Note: n is discrete-time

26. Extras • Note that there are two different ways to solve y(n). • You may use • y(n) = yh(n) + yp(n) or • y(n) = yzi(n) + yzs(n) • In either way, both method should yield the same solution y(n). You may refer the following link https://www.youtube.com/watch?v=isSFqzZwu38 • Also, note that the solution method for linear constant-coefficient difference equation for discrete-time systems is similar to that for second order differential equation, which we are often introduced.

27. Tutorial • Determine y(n) for the following difference equations • y(n) + a1y(n-1) = x(n) when x(n) = u(n) and y(-1) is the initial condition. • y(n) – 3y(n-1)-4y(n-2) = x(n) + 2x(n-1) when the input sequence is x(n) = 4nu(n).