Give an example of a complex signal of the form aet which, when sampled with sampling period 8 = 1, gives the values: 1, i, -1, -i, 1, i,-1,-i... (the sequence repeats). Notice that in (a), the four values assumed by the signal are the four 4th roots of unity. Given that the signal frequency is w, give a formula for a sample period & such that the sampled signal 1, s, iss takes values equal to the 4th roots of unity. Now generalize your answer to (b) for the case of arbitrary nth roots of unity. Given that the signal frequency is w, give a formula for & such that the sampled signal takes values equal to the nth roots of unity (your formula will be in terms of n and w). Give the smallest possible sampling period & such that the complex wave 2nift/N sampled with period & gives the N'th roots of unity. (Your formula for 8 will be in terms of f.)
Give an example of a complex signal of the form aet which, when sampled with sampling period 8 = 1, gives the values: 1, i, -1, -i, 1, i,-1,-i... (the sequence repeats). Notice that in (a), the four values assumed by the signal are the four 4th roots of unity. Given that the signal frequency is w, give a formula for a sample period & such that the sampled signal 1, s, iss takes values equal to the 4th roots of unity. Now generalize your answer to (b) for the case of arbitrary nth roots of unity. Given that the signal frequency is w, give a formula for & such that the sampled signal takes values equal to the nth roots of unity (your formula will be in terms of n and w). Give the smallest possible sampling period & such that the complex wave 2nift/N sampled with period & gives the N'th roots of unity. (Your formula for 8 will be in terms of f.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 3 part A-D please show step by step and explain
![**Exercise 2**
(a) Let \( z = a + bi \), and express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z, \overline{z}, e^{i\omega t}, \) and \( e^{-i\omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i\omega t} \) and \( e^{-i\omega t} \).
(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.
(c) In the chapter on complex numbers, we showed that for any complex number \( w \) the real part of \( w \) (written as \(\text{Re}[w]\)) can be expressed as:
\[
\text{Re}[w] = \frac{w + \overline{w}}{2}
\]
Use this fact to show that
\[ a \cos(\omega t) + b \sin(\omega t) = \text{Re}[z \cdot e^{i\omega t}] \]
As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves as the real part of complex signals composed of complex exponentials. It turns out that this enables us to bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely useful insights into the nature of these signals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F037991bd-d364-4fb8-a1f9-5eafda96396f%2Fo0174wp_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 2**
(a) Let \( z = a + bi \), and express \( a \cos(\omega t) + b \sin(\omega t) \) in terms of \( z, \overline{z}, e^{i\omega t}, \) and \( e^{-i\omega t} \). (Hint: Express \( \cos(\omega t) \) and \( \sin(\omega t) \) in terms of \( e^{i\omega t} \) and \( e^{-i\omega t} \).
(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.
(c) In the chapter on complex numbers, we showed that for any complex number \( w \) the real part of \( w \) (written as \(\text{Re}[w]\)) can be expressed as:
\[
\text{Re}[w] = \frac{w + \overline{w}}{2}
\]
Use this fact to show that
\[ a \cos(\omega t) + b \sin(\omega t) = \text{Re}[z \cdot e^{i\omega t}] \]
As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine waves as the real part of complex signals composed of complex exponentials. It turns out that this enables us to bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely useful insights into the nature of these signals.
Transcribed Image Text:**Digital Signals**
Most modern communications systems are digital. "Digital" means that in some sense the signal is associated with a discrete set of discrete values rather than a continuous range of values. So when the signal is received it's not measured continuously, but rather it is sampled by taking signal values at fixed regular intervals. The fixed interval between samples is called the sampling period, which we will denote by δ (Greek letter 'delta'). As a specific example, the complex signal \( e^{i\omega t} \) (here ω is a complex number) sampled with sampling period δ beginning at \( t = 0 \) gives the sequence of values: \( 1, e^{i\omega\delta}, e^{2i\omega\delta}, \dots \). In theory this is an infinite sequence, but it can repeat so that it only assumes a finite number of different values.
**Exercise 3.**
(a) Give an example of a complex signal of the form \( e^{i\omega t} \) which, when sampled with sampling period \( \delta = 1 \), gives the values: 1, i, -1, -i, 1, i, -1, -i, ... (the sequence repeats).
(b) Notice that in (a), the four values assumed by the signal are the four 4th roots of unity. Given that the signal frequency is \( \omega \), give a formula for a sample period δ such that the sampled signal \( 1, e^{i\omega\delta}, e^{2i\omega\delta}, \dots \) takes values equal to the 4th roots of unity.
(c) Now generalize your answer to (b) for the case of arbitrary nth roots of unity. Given that the signal frequency is \( \omega \), give a formula for δ such that the sampled signal takes values equal to the nth roots of unity (your formula will be in terms of n and ω).
(d) Give the smallest possible sampling period δ such that the complex wave \( e^{2\pi i f t / N} \) sampled with period δ gives the N'th roots of unity. (Your formula for δ will be in terms of f.)
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