Proving Magnitude & Phase of H(e^jw)

In summary, by rewriting H(e^jw) in different formats, we can see that the magnitude of H(e^jw) can be calculated by finding the real and imaginary parts of the complex exponential, and applying the definition of magnitude. This can be further simplified by using the property of complex conjugates, and ultimately we can find the magnitude and phase of H(e^jw). Additionally, we can explore if this is a general result or just a specific case by varying the amplitudes of the phasors.
  • #1
JeeebeZ
40
1

Homework Statement



H(e^jw) = (1-1.25e^(-jw))/(1-0.8e^(-jw))

Prove |H(e^jw)|^2 = G^2, and what is G
Find Magnitude & Phase

Homework Equations



H(e^jw) = (e^(jw)-1.25)/(e^(jw)-0.8)

H(e^jw) = 1 - (0.45e^(-jw))/(1-0.8e^(-jw))

H(e^jw) = 1 - 0.45/(e^(jw)-0.8)

The Attempt at a Solution



I don't know how to approch this question. I can rewrite H(e^jw) in 4 different formats, but none of which make me understand how to attempt to get the magnitude in the first place.
 
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  • #2
H(e^jw) = (1-1.25e^(-jw))/(1-0.8e^(-jw))
this would be:$$H(e^{j\omega})=\frac{1-(1.25)e^{-j\omega}}{1-(0.8)e^{-j\omega}}$$

What is the definition of the magnitude of a complex exponential/number?
 
  • #3
Simon Bridge said:
H(e^jw) = (1-1.25e^(-jw))/(1-0.8e^(-jw))
this would be:$$H(e^{j\omega})=\frac{1-(1.25)e^{-j\omega}}{1-(0.8)e^{-j\omega}}$$

What is the definition of the magnitude of a complex exponential/number?



$$Magnitude = \sqrt{Re(z)^2 + Im(z)^2}$$
 
  • #4
Great - so all you need to do is identify the real and imaginary parts of ##H(e^{j\omega})## ... how do you do that?
Hint: the exponential describes a phasor.
 
  • #5
Because |Z|^2 = z * zbar

we have
$$|H(e^{j\omega})|^2=(\frac{1-(1.25)e^{-j\omega}}{1-(0.8)e^{-j\omega}})(\frac{1-(1.25)e^{j\omega}}{1-(0.8)e^{j\omega}})$$

substituting
$$x=e^{j\omega}$$

$$|H(e^{j\omega})|^2=(\frac{1-(1.25)x^-1}{1-(0.8)x^-1})(\frac{1-(1.25)x}{1-(0.8)x})$$

$$|H(e^{j\omega})|^2=(\frac{1-(1.25)x^-1-(1.25)x + 1.25^2}{1-(0.8)x^-1-(0.8)x+0.8^2})(\frac{x}{x})$$

$$|H(e^{j\omega})|^2=\frac{x-(1.25)-(1.25)x^2 + 1.25^2x}{x-(0.8)-(0.8)x^2+0.8^2x}$$

$$|H(e^{j\omega})|^2=\frac{-1.25}{-0.80}\frac{x^2 + 2.05x - 1}{x^2 + 2.05x -1}$$

$$|H(e^{j\omega})|^2=\frac{-1.25}{-0.80} = 1.5625$$

$$|H(e^{j\omega})|= G = 1.25 = Magnatude$$

I got the Phase by computing the two angles, one on top with arctan (Im/Re), and the one on bottom, then phase total = top - bottom.
 
  • #6
See - you didn't need me :)
Hmmm ... I notice that 1.25 is the magnitude of one of the phasors in the combination which is a little startling. The second to last line says that the magnitude is the square-root of the ratio of the magnitudes of the phasors. You could explore to see if this is a general result or just a judicious choice of amplitudes.

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

Some latex notes (just sayin'):
you can use \left ( <some stuff> \right ) to fit brackets around the bigger stuff so
\left ( \frac{q}{p-1} \right ) gives you $$\left ( \frac{q}{p-1} \right )$$

the complex conjugate is usually better represented by a star notation as in ##z^*## or ##z^\star## - the bar gets tricky to typeset after a bit and you risk confusing ##z_{ave}=\bar{z}## ...
 
  • #7
I've never used latex before I just quoted yours and edited it as needed but I'll keep it in mind for the future. Thx
 
  • #8
You did well at that too :) Most people just go "meh".
Them I don't usually bother to give pointers ;)
 

Related to Proving Magnitude & Phase of H(e^jw)

1. What is the purpose of proving the magnitude and phase of H(e^jw)?

The purpose of proving the magnitude and phase of H(e^jw) is to analyze the frequency response of a system or signal. This allows us to understand how the system or signal behaves at different frequencies.

2. How is the magnitude of H(e^jw) calculated?

The magnitude of H(e^jw) is calculated by taking the absolute value of the complex transfer function. This gives us the amplitude of the output signal relative to the input signal at a specific frequency.

3. What does the phase of H(e^jw) represent?

The phase of H(e^jw) represents the shift in time between the input and output signals at a specific frequency. It can also indicate the amount of delay or advance in the output signal compared to the input signal.

4. How is the phase of H(e^jw) measured?

The phase of H(e^jw) is measured in degrees or radians. It can be calculated by taking the inverse tangent of the imaginary component divided by the real component of the complex transfer function at a specific frequency.

5. Why is it important to prove the magnitude and phase of H(e^jw)?

Proving the magnitude and phase of H(e^jw) allows us to understand the behavior of a system or signal at different frequencies. This information is crucial in designing and analyzing systems such as filters and amplifiers.

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