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.