数字信号处理（英文版）课后习题答案5

(Partial) Solutions to Assignment 5

pp.81-82

Discrete Fourier Series (DFS)

Discrete Fourier Transform (DFT)

, k=0,1,...N-1

, n=0,1,...N-1

Discrete Time Fourier Transform (DTFT)

is periodic with period=2π

Fourier Series (FS)

Fourier Transform (FT)

----------------------------------------------------

3.1 Using tthe definition, determine the DTFT of the following sequence (if it does not exist , give reason) (a) (c) (e) (a) ans:

(c) ans:

(e) ans:

Ans. (e). DTFT of

does not exist, because

diverge as

approaches ∞

---------------------------------------------------- 3.2 Given the fact that DTFT

the properties, compute the DTFT of each of the following sequences:

and using

(b) (d) b) ans:

Using the property:

Therefore

=

(d) ans:

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Q3.3 Compute the DFT of the following sequences: (a)

(c) x=[1,1,1,1,1,1,1,1]

ans: omitted.

---------------------------------------------------- Q 3.4 Let and

be the DFTs of two sequwences

respectively. Using properties of the DFT (do not compute the sequences), determine the

DFTs of the following (b) (d) (f)

(b) ans

Using the DFT property:

where

(d) ans

Using the the DFT property: Let

and we have

(d) and

Let

and we have

Therefore

where

---------------------------------------------------- 3.5

Let

,

be

an

8-point

sequence

with

DFT

. Using the propertiesof the DFT, determine the DFT of

the following sequences: a) b)

(a) ans DFT property:

where

and

,

with \

(b) ans

DFT property:

---------------------------------------------------- Q3.6 A 4-point real sequence (do not compute the sequence b) b) ans: DFT property:

where

and

has DFT

Using the properties of DFT

), determine the DFT of the following sequences

---------------------------------------------------- 3.7

the

finte

sequences,

and

, has DFT

Using the properties of the DFT, compute the following a) c)

a) ans:

Using DFT property:

c) ans:

where

denote circular convolution

---------------------------------------------------- Q 3.9. Let

be a finite sequence with DFT

Using the properties of DFT , determine the

following (a) (c)

a) ans: omitted c) ans: omitted

---------------------------------------------------- Q

3.10

Let

,with

Determine the DFT of the following sequences

(b)

and

b) ans:

=

Prove

Therefore

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