Exercise 2. Unlike real waves, complex waves can have negative frequencies. In terms of the circle picture just described, What is the difference between a complex wave with unit amplitude and frequency f, and a complex wave with unit amplitude and frequency - f? The real and imaginary parts of the signal with angular frequency w can have different amplitudes. This would correspond to a signal having the mathematical form a cos(wt) + ib sin(wt). It turns out to be convenient to express this signal in terms of complex exponentials instead of cosine and sine. Exercise 3. Given z = a + bi, express a cos(wt) + b sin(wt) in terms of z, et, and e-it. (Hint: Express cos(wt) and sin(wt) in terms of eft, and e-it. You will also need to use both z and teh complex conjugate 2 in your final expression.)
Exercise 2. Unlike real waves, complex waves can have negative frequencies. In terms of the circle picture just described, What is the difference between a complex wave with unit amplitude and frequency f, and a complex wave with unit amplitude and frequency - f? The real and imaginary parts of the signal with angular frequency w can have different amplitudes. This would correspond to a signal having the mathematical form a cos(wt) + ib sin(wt). It turns out to be convenient to express this signal in terms of complex exponentials instead of cosine and sine. Exercise 3. Given z = a + bi, express a cos(wt) + b sin(wt) in terms of z, et, and e-it. (Hint: Express cos(wt) and sin(wt) in terms of eft, and e-it. You will also need to use both z and teh complex conjugate 2 in your final expression.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do Exercise 2 and 3 and please show step by step and explain.
Transcribed Image Text:**Complex Waves in Signal Processing**
Modern electronic communication systems rely on signals based on sinusoidal waves. Two primary examples of sinusoidal waves (as functions of time) are \(\cos(2\pi ft)\) and \(\sin(2\pi ft)\). Here, \(f\) represents the **frequency**, typically measured in cycles per second (Hertz, or 'Hz'). For example, a frequency of 66 Hz indicates that the signal makes 66 complete cycles in one second.
A related quantity is the **angular frequency**, usually denoted by the Greek letter \(\omega\) ('omega'). The angular frequency \(\omega\) is measured in radians per second, and \(\omega\) and \(f\) are related by the formula \(\omega = 2\pi f\).
Notice that \(\cos(2\pi ft)\) and \(\sin(2\pi ft)\) are the real and imaginary parts of the function \(e^{i(2\pi ft)}\) (or \(e^{i\theta}\), as stated earlier). In many wireless communications systems, signals are transmitted in pairs, where one signal is attached to a cosine wave and the other is attached to the corresponding sine wave. These two paired signals are denoted by "in phase" and "quadrature phase"; we can conveniently represent these as the "real part" and the "imaginary part" of a single complex wave.
This explanation transitions into the use of \(e^{i\theta}\) notation, which is prevalent in advanced mathematics, physics, and engineering references.
).
**Exercise 2.** Unlike real waves, complex waves can have negative frequencies. In terms of the circle picture just described, what is the difference between a complex wave with unit amplitude and frequency \( f \), and a complex wave with unit amplitude and frequency \( -f \)?
_The real and imaginary parts of the signal with angular frequency \( \omega \) can have different amplitudes. This would correspond to a signal having the mathematical form \( a \cos(\omega t) + ib \sin(\omega t) \). It turns out to be convenient to express this signal in terms of complex exponentials instead of cosine and sine._
**Exercise 3.** Given \( z = a + bi \), express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z \), \( e^{i \omega t} \), and \( e^{-i \omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i \omega t} \), and \( e^{-i \omega t} \). You will also need to use both \( z \) and the complex conjugate \( \bar{z} \) in your final expression.)
---
There are no graphs or diagrams to describe.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F46d6be7d-d275-4947-99a2-8d175e979628%2Fkfl1nif_processed.png&w=3840&q=75)
Transcribed Image Text:Here's the transcription of the image text for an educational website:
---
Amplitude can be visualized as a point travelling around the unit circle in the complex plane (see [YouTube video](https://www.youtube.com/watch?v=ZnZHdta97K4)).
**Exercise 2.** Unlike real waves, complex waves can have negative frequencies. In terms of the circle picture just described, what is the difference between a complex wave with unit amplitude and frequency \( f \), and a complex wave with unit amplitude and frequency \( -f \)?
_The real and imaginary parts of the signal with angular frequency \( \omega \) can have different amplitudes. This would correspond to a signal having the mathematical form \( a \cos(\omega t) + ib \sin(\omega t) \). It turns out to be convenient to express this signal in terms of complex exponentials instead of cosine and sine._
**Exercise 3.** Given \( z = a + bi \), express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z \), \( e^{i \omega t} \), and \( e^{-i \omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i \omega t} \), and \( e^{-i \omega t} \). You will also need to use both \( z \) and the complex conjugate \( \bar{z} \) in your final expression.)
---
There are no graphs or diagrams to describe.
Expert Solution
Step 1
We are given the following.
The notation instead of .
Exercise 2:
Unlike real waves, complex waves can have negative frequencies.
As per the described circle picture, complex wave with unit amplitude and frequency f is defined as .
And as already explained before, we have .
Now, a complex wave with unit amplitude and frequency -f is defined as the following.
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
