数字信号处理习题汇编(1)

习题汇编

第一部分：信号与系统的时域分析

1. Concepts:

1.1 The unit impulse response and the linear convolution

The unit impulse response h[n] of a discrete-time L TI system is the system response to the unit impulse sequence δ[n] when the system has no initial energy.

The concept of the impulse response is very very important. In the time-domain, an L TI system can be uniquely characterized by its impulse response h[n], so, we often use the unit impulse response to represent an L TI system. In this case, the input-output relationship of an L TI system is described by the convolution operation:

∑∞-∞=-

=

k

k

n

h

k

x

n

y]

[

]

[

]

[

Physical meaning: The convolution sum operation has an explicit meaning, which is used to determine the system response. If the system is determined, then its impulse response is unique. We can compute the system response to arbitrary input signals.

Note: The key to compute the convolution sum is determination of the summation interval(求和区间). So we often need to graph x[k] and h[h-k]. Using the graphs of x[k] and h[h-k] can help us to determine the summation interval readily.

Requirements:

Understand the concept of the impulse response, be able to calculate the

system response using the convolution sum operation.

Exercises:

Compute the linear convolution y[n]=x[n]*h[n] of the following pairs of signals:

(a) ]

[][][][n n h n n x n

n

μβμα== β

α

≠

(b) ]

[][][]

[][N n n n h n n x n

--==μμμα N is a positive integer

(b) ]5[][][}6,5,4,3,2,1{][--==n n n h n x μμ (c)

]

3[]2[][]

[5.0][--+==n n n h n n x n

μμμ

Answer the following questions:

(a) How to determine the range of the index n for which the system response y[n] is defined?

(b) How to determine the length of the system response y[n] which is computed from the convolution sum?

1.2 The linear constant-coefficient difference equations and their solutions The convolution sum is one of the mathematical models of the L TI system. In general, we can use a difference equation to represent the input-output relationship of an L TI system.

∑∑==-=

-M

k k

N

k k

k n x b

k n y a

][][

The difference equation gives an implicit relation about y[n]. In order to obtain an explicit expression of y[n], we must solve the equation.

Procedure:

(a) Determine the characteristic equation from the difference equation:

00

=∑=N

k n

k

a

λ

(b) Determine the characteristic roots, then the homogeneous solution can be determined (if no repeated roots):

n

N

N n

n

c C C C n y λλλ+++= 2211][

(c) Assume the particular solution having the same form to the input signal, for example, if the input is x[n] = αn μ[n], we can assume that the particular solution to be y p [n] = K αn μ[n], substituting y p [n] into the difference equation we can determine the constant K.

(d) The complete solution is then given by

][][][n y n y n y p c +=

(e) Translate the initial conditions y[-1], y[-2], … , y[-N] to obtain a set of new additional conditions y[0], y[1], … ,y[N-1] by the recursive method. Substituting y[0], y[1],…,y[N-1] in to the complete solution y[n], then the coefficients C 1, C 2, … , C N can be determined.

Exercises:

Solving the following difference equations (a)

]

[]1[4

1][n x n y n y =--

with the input x[n] = (0.5)n μ[n] and the initial

condition y[-1] = 8

(b)

][]2[8

1]1[4

1][n x n y n y n y =--

--

with the input x[n] = (0.2)n μ[n] and the

initial condition y[-1] = -1, y[-2] = 1.

(c) ]1[2][]2[06.0]1[1.0][--=---n x n x n y n y n y ＋ with the initial conditions y[-1]

= -1, y[-2] = 2, and the input x[n] = (0.4)n μ[n].

(d) The sequence of Fibonacci numbers f[n] is a causal sequence defined by

,2],

2[]1[][≥-+-=n n f n f n f

with f[0]=0 and f[1]=1.

(1) Develop an exact formula to calculate f[n] directly for any n. (2) Show that f[n] is the impulse response of a causal L TI system described by the difference equition

]1[]2[]1[][-+-+-=n x n y n y n y

Solution for (c):

We first determine the homogenous solution:

The characteristic equation: 006.01.02

=-λλ＋

The roots are: 3.01-=λ

and

2.01=λ

The homogeneous solution is n

n

c C C n y )

2.0()

3.0(][21+-= for n>0

Determine the particular solution

Assume that the particular solution is n

p K n y )

4.0(][=. Obviousely, ][n y p

satisfies the difference equation, substituting ][n y p

into the difference equation

we arrive at

]1[)

4.0(2][)4.0()

4.0(06.0)

4.0(1.0)4.0(1

2

1

--=----n n K K K n n n n n

μμ＋

For n ≥ 1, the above equation is

n

n n n n

K K K )

4.0(4]4

.021[)4.0()

4.0(06.0)4.0(1.0)4.0(1

2

1

-=?-=----＋

i.e.,

4)

4.0(06.0)

4.0(1.02

1

-=---K K K ＋ →

3

16-

=K

Then the particular solution is n

p n y )

4.0(3

16][－

=

The total solution is then

n

n

n

C C n y )

4.0(3

16)2.0()3.0(][21-

+-=

By the use of the initial conditions we get 22.1]1[2]0[]2[06.0]1[1.0]0[=----=x x y y y ＋＋－

782

.1]0[2]1[]1[06.0]0[1.0]1[-=--=x x y y y ＋＋－

Substituting the above into the total solution we obtain the following equations:

22

.1316]0[21=-

+=C C y

782

.1)4.0(3

16)2.0()3.0(]1[21-=-

+-=C C y

So

63

.492.121==C C

The total solution is

n

n

n

n y )

4.0(3

16)2.0(63.4)3.0(92.1][-

+-= n ≥ 0

1.3 The properties of L TI discrete-time systems (a) Linearity (b) Time-invariance (c) BIBO stability

If bounded input leads to a bounded output, then the system is said to be the BIBO stable.

Because we often use the impulse response h[n] to represent an L TI system, so we can use the following condition to determine the stability of an L TI system:

∞

<∑

∞

-∞=n n h ][ This condition is called the absolutely summable condition .

(d) Causality

A system is said to be causal if the present output y[n] at n = n 0 depends only on the present input x[n 0] and the past x[n], n

In other word, if the impulse response h[n] is causal, then the system is causal.

Requirements :

Understand the above properties of the L TI system. Given a mathematic model of an L TI system, you should be able to determine the properties that the system possesses.

Exercises:

For each of the following discrete-time systems, where y[n] and x[n] are, respectively, the output and the input sequences, determine whether or not the system is (1) linear , (2) causal, (3) stable, and (4) shift-invariant.

(a)

];[][3

n x n n y =

(b) ;])[(][5n x n y =

(c)

];2[][+-=n x n y αα

is a nonzero constant

Direction:Fill the best answer into the bracket for each of the following sentences.

1. A causal LTI discrete-time system is described by the difference equation:

]2[2

1]1[][2

1][-+

-+=

n x n x n x n y , its unit impulse response h[n] = (

{2

1,

1,2

1} ).

说明：这是一个FIR 系统，对于FIR 系统，其差分方程的右边系数，即为系统的单位冲激响应。

2. The fundamental period of the discrete-time sequence x[n] = cos(0.4πn) is ( 5 ). 说明：基本周期的定义即计算公式：k

N ω

π

2=

，其中N 和k 均为整数，N 为基本周期

（使得N 为最小整数时k 取值）。本题ω = 0.4π，代入上式得到：1,

5==k N 。

3. Given two sequences x[n] and h[n], their lengths are N and M, respectively, then the length of their linear convolution is ( ).

4. The relationship between the unit impulse sequence and the unit step sequence is ( ).

5. An LTI discrete-time system with its unit impulse response h[n] = δ[n-2], then the

output response y[n] to the input sequence x[n] is equal to ( ).

6. Assume that the unit step response of an LTI discrete-time system is s[n], the unit impulse response h[n] can be expressed as ( ).

7. Sampling the continuous-time signal x(t) = sin(200πt) at sampling frequency F T = 50 Hz, we can obtain a discrete-time sequence x[n] = ( ).

8. The unit impulse response of an LTI discrete-time system is h[n] = 0.5n+1μ[n+1], then the system is ( ). (Stability, causality)

湛柏明

2012-12-05