Reshaping Complex Equations in LTI-Systems: Solving for the System Function H(z)

[mkkribor]

Hi!

New go on my problem with LTI-system which really is a math-problem:

My problem i can't solve:

Compute the System Function H(z) from the Frequency

H(e^jw)=2*exp^(-j*3/2*w)*[cos(w/2)]^2

In other words i need to reshape the equation where i only have exp^(-j*k*w), where k is an integer.

Then the answer should be in the form:
H(z)= (b_0+b_1 z^(-1)+b_2 z^(-2)+⋯)/(1-a_1 z^(-1)-a_2 z^(-2)-…)
where z = exp^jw, and the b and a are constants.

Thx for any help!

 

[tiny-tim]

welcome to pf!

hi mkkribor! welcome to pf!

(try using the X² icon just above the Reply box )

hint: use Euler's equation and the formula x^ab = (x^a)^b

 

[mkkribor]

Thx for answer, and I see that solves it for what I first asked for. But then i get squareroots and I still can't get it in the form I want:

H([e]^{}[/jw]) = [_{}[/0]+_{}[/1][e]^{}[/-jw]+...+_{}[/m][e]^{}[/-jmw]]\frac{}{}[/1-[a]_{}[/1][e]^{}[/-jw]+...+[a]_{}[/n]*[e]^{}[/-jnw]]

where n,m = integers and the a-s and b-s can be anything. (Most likely in this case, its an FIR-filter, which means that all the a-s coefficients are zero)

Can you help me on this please?

 

[mkkribor]

Ok, i clearly don't understand how to use the formating;), but i can attach a photo of the equation if you doent see what it says?

 

[tiny-tim]

hi mkkribor!

hmm … let's decode this …

H(e^/jw) = b₀ + b₁e^/jw + ... + _{}[/m]e^-/jw\frac{}{}[/1-[a]_{}[/1][e]^{}[/-jw]+...+[a]_{}[/n]*[e]^{}[/-jnw]​

no, i give up

to make b₁e^jw, type [NOPARSE]"b₁e^jw"[/NOPARSE]