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MATH 640 Homework: Filter Description, Invertibility, and Commutation, Assignments of Mathematics

Drexel University Mathematics

The winter 2002/2003 homework assignment for math 640, due on january 30, 2003. The assignment includes problems related to filter description, invertibility, and commutation. Students are required to describe the operators h1, h2, and h3, find their z-transforms, determine when these filters are invertible, and show that every filter commutes with every other filter. Additionally, students must find non-zero inputs, frequencies, and numbers that result in zero outputs for specific filters.

Typology: Assignments

Pre 2010

Uploaded on 08/19/2009

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Winter 2002/2003 Homework MATH 640

DUE: January 30, 2003

1. (a) Describe the operator



2





2





2





2



and its z-transform.

(b) Describe the operator



3



and



3



and



3





3



and its z-trn

2. When are these filters invertible? Which has a causal inverse? Which has an FIR inverse?

(a) H1



z





1



αz





1



βz1



.

(b) H2



z





z



β





1



βz1



.

(c) H3



z



1



βz1



z2.

3. Show that every filter commutes with every other filter. Recall that a filter has the form ∑h



n



Snwhere Sis the

delay operator.

4. Show that H



S



S1is not invertible in three ways. Find a non-zero input xso that Hx



0. Find a frequency

ωthat has response H



eiω



0. Find a number with



z



1 such that H



z



0.

5. What are the matrix Hand coefficient vector hfor the 3-term moving average Hx



n





1



3





x



n





x



n



1





x



n



2



? This is not invertible. Find two vectors xfor which Hx



0. Find two numbers with



z



1 such that

H



z



0. Find two frequencies such that H



ω



0.

1

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Winter 2002/2003 Homework MATH 640 DUE: January 30, 2003

1. (a) Describe the operator ^2 ^2 ^2 ^2 and its z -transform.

(b) Describe the operator ^3 and ^3 and ^3 ^3 and its z -trn

When are these filters invertible? Which has a causal inverse? Which has an FIR inverse?

(a) H 1 z 1 α z 1 β z^1 ^.

(b) H 2 z z β 1 β z^1 ^.

(c) H 3 z 1 β z^1 z^2.

3. Show that every filter commutes with every other filter. Recall that a filter has the form ∑ h n Sn^ where S is the

delay operator.

MATH 640 Homework: Filter Description, Invertibility, and Commutation, Assignments of Mathematics

Related documents

Partial preview of the text

Download MATH 640 Homework: Filter Description, Invertibility, and Commutation and more Assignments Mathematics in PDF only on Docsity!

1. (a) Describe the operator ^2 ^2 ^2 ^2 and its z -transform.

(b) Describe the operator ^3 and ^3 and ^3 ^3 and its z -trn

(a) H 1 z 1 α z 1 β z^1 ^.

(b) H 2 z z β 1 β z^1 ^.

(c) H 3 z 1 β z^1 z^2.

3. Show that every filter commutes with every other filter. Recall that a filter has the form ∑ h n Sn^ where S is the

4. Show that H S S^1 is not invertible in three ways. Find a non-zero input x so that Hx 0. Find a frequency

ω that has response H ei ω^ 0. Find a number with z 1 such that H z 0.

5. What are the matrix H and coefficient vector h for the 3-term moving average Hx n 1 3 x n x n 1

x n 2 ? This is not invertible. Find two vectors x for which Hx 0. Find two numbers with z 1 such that

H z 0. Find two frequencies such that H ω 0.

MATH 640 Homework: Filter Description, Invertibility, and Commutation, Assignments of Mathematics

Related documents

Partial preview of the text

Download MATH 640 Homework: Filter Description, Invertibility, and Commutation and more Assignments Mathematics in PDF only on Docsity!

1. (a) Describe the operator ^2 ^2 ^2 ^2 and its z -transform.

(b) Describe the operator ^3 and ^3 and ^3 ^3 and its z -trn

(a) H 1 z  1  α z 1  β z^1 ^.

(b) H 2 z  z  β  1  β z^1 ^.

(c) H 3 z  1  β z^1 z^2.

3. Show that every filter commutes with every other filter. Recall that a filter has the form ∑ h n Sn^ where S is the

4. Show that H  S S^1 is not invertible in three ways. Find a non-zero input x so that Hx  0. Find a frequency

ω that has response H ei ω^  0. Find a number with z  1 such that H z  0.

5. What are the matrix H and coefficient vector h for the 3-term moving average Hx n  1  3 x n x n  1

x n  2 ? This is not invertible. Find two vectors x for which Hx  0. Find two numbers with z  1 such that

H z  0. Find two frequencies such that H ω  0.

(a) H 1 z 1 α z 1 β z^1 ^.

(b) H 2 z z β 1 β z^1 ^.

(c) H 3 z 1 β z^1 z^2.

4. Show that H S S^1 is not invertible in three ways. Find a non-zero input x so that Hx 0. Find a frequency

ω that has response H ei ω^ 0. Find a number with z 1 such that H z 0.

5. What are the matrix H and coefficient vector h for the 3-term moving average Hx n 1 3 x n x n 1

x n 2 ? This is not invertible. Find two vectors x for which Hx 0. Find two numbers with z 1 such that

H z 0. Find two frequencies such that H ω 0.