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 integer N≠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.

kN

kωN

nNnn

2

2

)),(cos()cos(

Z

3

Review: characteristics of discrete-time systemThe 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)},thenH { 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.

n

nh )(

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.

)()()()()( nhnxknhkxnyk

)]()([)()( 21 nhnhnxny

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.

M

l

l lnxbny0

)()(

M

l

l

N

i

i lnxbinyany01

)()()(

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.

n

nznxnxzX ][]}[{)( Z

7

Review: Convergence of the z transform

21

2121

02

02

01

01

0

1

10

][)(

0 0

0 0 ][)( 左边边序

0

0 ][)( 右边边序

0 0

0 0

0,0n 0

][)(

0

0

1

0

rr

rrrzrznxzX

nrz

nrzznxzX

nzr

nzrznxzX

nz

nz

nz

znxzX

n

n

n

n

n

nn

n

n

nn

n

无收敛域收敛域为双边序列

收敛域为

收敛域为

收敛域为有限长序列

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. 11 r

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).

)(

)()(

zD

zNzX

10

Review: PolesX(z) is also can be expressed as

K

k

mk

kpz

zNzX

1

)(

)()( 因为 X(z) 在极点处无

意义，所以其收敛域一定不包括极点。

因为 X(z) 在极点处无意义，所以其收敛域一定不包括极点。

• 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

Review: Figure 2.2 (a)

r1

Im{z}

Re{z}

12

Review: Figure 2.2 (b)

r2

Im{z}

Re{z}

13

Review: Figure 2.2 (c)

r1r2

Re{z}

Im{z}

14

Review: The z transform of basic sequences

x(n) X(z) ROC

1

u(n)

anu(n)

anu(n-1)

nu(n)

nanu(n)

)(n z0

1z

z z1

az

z

za

az

a

2)1( z

z z1

2)( az

az

za

za

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 .

K

k

m

ii

k

kiLM

l

ll

k

pz

czgzX

1 10 )()(

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

k

k

k

k

pz

mkm

m

kki zXpz

dz

d

mc

)]()[()!1(

1)1(

)1(

kpzkki zXpzc

)()(

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].

2102 ]2[]1[]0[]1[]2[

][)(

zxzxzxzxzx

znxzXn

n

18

Review: Time-shift theoremTime-shift theorem （时移定理）

Assume that x[n] ↔ X(z), then x[n + l] ↔ zl

X(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

M

ll

N

ii lnxbinya

00

][][

}][{]}[{00

M

ll

N

ii lnxbinya ZZ

M

l

ll

N

i

ii zXzbzYza

00

)()(

20

Review: Transfer functionsMaking 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).

N

i

ii

M

l

ll

za

zb

zX

zYzH

1

0

1)(

)()(

][][][)()()( nhnxnyzHzXzY

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].∈

deeXnx

enxeX

njj

n

njj

)(2

1][

][)(

22

Review: Properties of the Fourier transform

Several properties:x(n) X(ejω)

realreal

imaginaryimaginary

realreal

imaginaryimaginary

conjugatesymmetric

conjugatesymmetric

conjugate antisymmetri

c

conjugate antisymmetri

c

conjugatesymmetric

conjugatesymmetric

conjugate antisymmetric

conjugate antisymmetric

23

Review: Frequency response

H(z)（系统函数）

H(ejω)（频率响应）

jezk

kjj zHekhnheH

)()()}({)( F

( ) Z{ ( )} ( ) k

k

H z h n h k z

h(n) — 单位冲

激响应

)()(

)()()()()()( )()(

jnj

k

kjnj

k

knjenx

k

eHeekhe

khekhknxnhnxnynj

－ 复正弦输入得到复正弦输出

)()(

)()()()()()()(

zHzzkhz

khzkhknxnhnxny

n

k

kn

k

knznx

k

n

－ 指数输入得到指数输出

Quiz OneMarch 21th, 2011

25

1.Characterize the system below as linear/nonlinear, causal/noncausal and time invariant/time varying.

y(n)=(n+a)2x(n+4)

2.For the following discrete signal, determine whether it’s periodic or not. Calculate the fundmental period if it is periodic.

2 2( ) cos ( )

15x n n

26

3. Compute the convolution sum of the following pairs of sequences.

4. Discuss the stability of the system described by the impulse response as below:

h(n)=0.5nu(n)-0.5nu(4-n)

therwise , 0

41 , )( and

therwise , 0

87 , 1

63 , 0

20 , 1

)(o

nnnh

o

n

n

n

nx

5. Compute the Fourier transform of the following sequences:

6. Given x(n) as following, find X(z) and discuss its ROC.

( ) ( )nx n a u n

]2[3][2 nunx n

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

]1[]2[]1[][ nxnynyny