Chapter 2 Answers

2.1. (a) We know that

0 0

yi[n] = x[n] * h[n] = ̂ h[k]x[nk = - o o

- k ] (S2.1-1)

The signals x[n] and /i[n] are as shown in Figxire S2.1.

, - t WA

2 .4 h W

3

0 \ t ^ - 1 0i )

1

Figure S2.1

Prom this figure, we can easily see that the above convolution sum reduces to

yi[n] = /i[-l]a:[n + l] + /i[l]a:[n= 2x[n + 1] + 2x[n — 1

- 1 ]

This gives

yi [n] = 25[n + 1] + A5[n] + 2<5[n — 1] + 25[n --2] - 1<M1

(b) We know thatO O

y2[n] = x[n + 2] * /i[n] = ̂ h[k]x[nk — - o o

Comparing with eq. (S2.1-1), we see that

+ 2 -- k ]

y2[n] = yi[n + 2;

(c) We may rewrite eq. (S2.1-1) as0 0

yi[n] = x[n] * /i[n] = ̂ x[k]h[nk = - o o

- k ]

Similarly, we may write

0 0

yz[n] = a;[n] * h[n + 2] = ̂ i[/i:]/i[nf c = - o o

+ 2 -- k ]

Comparing this with eq. (S2.1), we see that•

!/3W = yi[n + 2]

30

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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their c

ourse

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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e use

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tors i

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ourse

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will de

stroy

the i

ntegri

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the w

ork an

d is n

ot pe

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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will de

stroy

the i

ntegri

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the w

ork an

d is n

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

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ided s

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e use

of in

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tors i

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ourse

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asse

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will de

stroy

the i

ntegri

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the w

ork an

d is n

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

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ided s

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of in

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tors i

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will de

stroy

the i

ntegri

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the w

ork an

d is n

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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e use

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tors i

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ourse

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asse

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will de

stroy

the i

ntegri

ty of

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d is n

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

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e use

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tors i

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ourse

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asse

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stud

ent le

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ide W

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

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their c

ourse

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asse

ssing

stud

ent le

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

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. Diss

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tion

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work (in

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g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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ided s

olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

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. Diss

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tion

or sa

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any p

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work (in

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g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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ided s

olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

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. Diss

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tion

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work (in

cludin

g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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ided s

olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

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. Diss

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tion

or sa

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work (in

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g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

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struc

tors i

n tea

ching

their c

ourse

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asse

ssing

stud

ent le

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. Diss

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tion

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work (in

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g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

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. Diss

emina

tion

or sa

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any p

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work (in

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g on t

he W

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ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

arning

. Diss

emina

tion

or sa

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any p

art of

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work (in

cludin

g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

s and

asse

ssing

stud

ent le

arning

. Diss

emina

tion

or sa

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any p

art of

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work (in

cludin

g on t

he W

orld W

ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

of in

struc

tors i

n tea

ching

their c

ourse

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asse

ssing

stud

ent le

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. Diss

emina

tion

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work (in

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g on t

he W

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ide W

eb)

will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

for th

e use

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tors i

n tea

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their c

ourse

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asse

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ent le

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. Diss

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g on t

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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olely

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e use

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tors i

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ourse

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asse

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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for th

e use

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tors i

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ourse

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asse

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ent le

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. Diss

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tion

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g on t

he W

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ide W

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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e use

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tors i

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ourse

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asse

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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e use

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tors i

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ourse

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asse

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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for th

e use

of in

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tors i

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ourse

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asse

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. Diss

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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e use

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tors i

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ourse

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asse

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. Diss

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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tors i

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asse

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will de

stroy

the i

ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

d.

© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

This w

ork is

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tors i

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will de

stroy

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ntegri

ty of

the w

ork an

d is n

ot pe

rmitte

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© 1997 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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ork is

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tors i

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