Review for chapter two & three

1. Review for chapter two & three

2. Review :periodicity of sequence • A sequence x[n] is defined to be periodic if and only if there is an integerN≠0 such that x[n] = x[n+N] for all n. In such a case, N is called the period of the sequence. • Note, not all discrete cosine functions are periodic. • If 2π/ω is an integer（整数） or a rational number（有理数), this sequence will be periodic; • If 2π/ω is an irrational number（无理数）, this cosine function will not be periodic at all.

3. Review: characteristics of discrete-time system The characteristics of the discrete-time system y(n) = H {x(n)} : • Linearity: If y1(n)= H { x1(n)}, y2(n)= H { x2(n)},then H {ax(n)}=aH {x(n)} and H { x1(n)+ x2(n)}=H {x1(n)}+H {x2(n)} for any constants a and b. • time invariance:If y (n)= H { x(n)},then H { x(n-n0)}=y(n-n0) • Causality: If, when x1(n) = x2(n) for n < n0, then H {x1(n)} = H {x2(n)}, for n < n0 • Stability: For every input limited in amplitude, the output signal is also limited in amplitude.

4. Review: LTI system • The output y(n) of a linear time-invariant system can be expressed as where h(n) = H {δ(n)} is the impulse response of the system. • Two linear time-invariant systems in cascade form a linear time-invariant system with an impulse response which is the convolution sum of the two impulse responses.

5. Review: FIR & IIR systems • A nonrecursive system such as are often referred to as finite-duration impulse-response (FIR) filters. • A recursive digital system such as (at least one ai≠0)are often referred to as infinite-duration impulse-response (IIR) filters.

6. Review: z-transform • The z transform X(z) of a sequence x(n) is defined as where z is a complex variable. The z transform given by this equation is referred to as the two-sided z transform.

7. Review: Convergence of the z transform

8. Review: H(z) for a stable causal system • For a causal system, its ROC should be |z|>r1. • For a stable system, its ROC should include unit circle. • So the convergence circle’s radius of the z transform of the impulse response of a stable causal system should be smaller than unit. i.e.

9. Review: Zeros and poles • An important class of z transforms are those for which X(z) is a ratio of polynomials in z, that is • The roots of the numerator polynomial （分子多项式）N(z) are those values of z for which X(z) is zero and are referred to as the zeros（零点） of X(z). • Values of z for which X(z) is infinite are referred to as the poles （极点）of X(z). The poles of X(z) are the roots of the denominator polynomial（分母多项式） D(z).

10. Review: Poles • 因为X(z)在极点处无意义，所以其收敛域一定不包括极点。 X(z) is also can be expressed as • Right-handed: The convergence region is |z| > r1, so all the poles must be inside the circle |z| = r1, r1 = max{|pk|}(figure (a); • Left-handed: The convergence region is |z| < r2, all the poles must be outside the circle |z| = r2, r2 = min{|pk|} (figure (b); • Two-sided sequences: The convergence region is r1 < |z| < r2, some poles are inside the circle |z| = r1 and the others outside the circle |z| = r2 (figure (c).

11. Im{z} r1 Review: Figure 2.2 (a) Re{z}

12. Im{z} r2 Review: Figure 2.2 (b) Re{z}

13. Im{z} r2 Re{z} r1 Review: Figure 2.2 (c)

14. Review: The z transform of basic sequences

15. Review: inverse z transform • For rational z transform, a partial-fraction expansion is carried out firstly, and then the inverse z transforms of the simple terms are identified. • If X(z) = N(z) / D(z) has K different poles pk, k = 1,…,K, each of multiplicity mk, then the partial-fraction expansion（部分分式展开法）of X(z) is where M and L are the degrees of the numerator and denominator of X(z), respectively. • The coefficients gl, l = 0,…, M – L, is the quotient（商） of polynomials N(z) and D(z). If M < L, then gl = 0 for any l .

16. Review: Partial-fraction expansion • The coefficients cki are given by Particularly, in the case of a simple pole（单极点）, cki is given by Since the z transform is linear and the inverse z transform of each of the terms is easy to compute, then the inverse z transform follows directly from the above equation

17. Review: long division • For X(z) = N(z) / D(z), we can perform dividing N(z) by D(z) and the quotient商 is a power series of z. • In the power series幂级数, the coefficient of the term involving z–n simply corresponds the sequence x[n].

18. Review: Time-shift theorem Time-shift theorem（时移定理） Assume that x[n] ↔ X(z), then x[n + l] ↔ zlX(z), where l is an integer. If the ROC of X(z) is r1 < |z| < r2, then the ROC of Z{x(n+l)} is the same as the ROC of X(z). If x[n] is right-handed or left hand, the ROC of Z{x[n+l]} is the same as the ROC of X(z), except for the possible inclusion or exclusion of the regions z=0 and |z| = ∞.

19. Review: Transform-Domain Analysis of LTI Discrete-Time System • A linear system can be characterized by a difference equation as follows Applying the z transform on both sides, we get that Applying the time-shift theorem, we obtain

20. Review: Transfer functions Making a0=1, we define as the transfer function of the system relating the output Y(z) to the input X(z). • From the convolution theorem, we have therefore the transfer function of the system is the z transform of its impulse response h(n).

21. Review: Frequency-domain representation of discrete-time signals and systems • The direct and inverse Fourier transforms of the discrete-time signal x(n) are defined as • In fact, the Fourier transform X(ejω) is the z transform of the discrete-time signal x(n) at the unit circle. • the Fourier transform X(ejω) is periodic with period 2π, therefore the Fourier transform of x(n) requires specification only for a range of 2π, for example, ω∈[-π,π] or ω ∈[0, 2π].

22. Review: Properties of the Fourier transform Several properties: x(n) X(ejω) real real imaginary imaginary conjugate symmetric conjugate symmetric conjugate antisymmetric conjugate antisymmetric

23. － 指数输入得到指数输出 h(n) — 单位冲激响应 － 复正弦输入得到复正弦输出 Review: Frequency response H(z) （系统函数） H(ejω) （频率响应）

24. Quiz One March 21th, 2011

25. Characterize the system below as linear/nonlinear, causal/noncausal and time invariant/time varying. y(n)=(n+a)2x(n+4) • For the following discrete signal, determine whether it’s periodic or not. Calculate the fundmental period if it is periodic.

26. Compute the convolution sum of the following pairs of sequences. • Discuss the stability of the system described by the impulse response as below: h(n)=0.5nu(n)-0.5nu(4-n)

27. Compute the Fourier transform of the following sequences: 6. Given x(n) as following, find X(z) and discuss its ROC.

28. 7. A LTI causal system can be described by the different equation: 1) Compute the transfer function H(z) of the system. 2) Compute the impulse response h[n] of system. 3) compute the frequency response H(ejw) of system. 4)Determine the system is stable or not. tips: if ROC of H(z) includes |z|=1