Sinusoids as Complex numbers (multiplication query)

[Natalie Johnson]

DSP Guide .com has the highly rated textbook for digital signal processing.
Chapter 30 pg 561 on Complex Numbers
http://www.dspguide.com/ch30.htm (chapters are free to download)

Hes talking about representing sinusoids with a complex number.

Author states "Multiplying complex numbers A and B, results in another complex number". "Multiplying two sinusoids does not produce another sinusoid. Complex multiplication fails to match the physical system".

But then the following page 562
"When a sinusoid passes through a linear system, the complex numbers representing the input signal and the system are multiplied, producing the complex number representing the output".

Which is contrary to what he stated? The final complex number does actually give a sinusoid after multiplication, so matches the physical system...

What am I missing here?

 

[fresh_42]

The pass through a linear system is the difference at first glimpse. So without further context, I assume it is this, because multiplication with a fixed number is also linear.

 

[scottdave]

Is a linear system a sinusoid, or something else? Think about that.
But you can multiply 2 sinusoids and get another sinusoid. The frequency may be different, though.

 

[Natalie Johnson]

Is a linear system a sinusoid, or something else? Think about that.
But you can multiply 2 sinusoids and get another sinusoid. The frequency may be different, though.

Yes the linear system is being represented as a complex sinusoid

 

[BvU]

Ah, so the book title is "The Scientist and Engineer's Guide to Digital Signal Processing" and not

'DSP for Engineers and Scientists'

Big fat book ! But the website says " Available soon! " -- ah, there is some PDF

On p 561 he explains that addition and subtraction are OK (because real and imaginary parts are not mixed).
For multiplication things do mix up in the way he presents it.

I am very unhappy with the way he treats this material -- and clearly confuses you.
It already goes badly wrong on p 560. It is nonsense to equivalence a real function of $\omega t$ to a complex number $a+jb$It's beyond the PF scope to rewrite his book, but the way it should be done is to present the real sinusoid $$A\cos(\omega t)+ B\sin(\omega t) = M \cos(\omega t + \phi) $$ as the real part of the complex number (function of M, $\phi, \omega$) $$M e^{j\omega t + \phi} $$ (with M, the amplitude, $\phi$ and $\omega$ all real numbers)

This way complex multiplication and all other kinds of manipulation work just fine, something he happily uses on p 562.

Which is contrary to what he stated?

You are right: Yes, it is. And it confuses readers no end. We should send him a link to your post as feedback -- perhaps he can improve on it ...

'Highly rated' it may be, but I'd go look for a better book / pdf ) There's plenty around.

 

[Natalie Johnson]

Ah, so the book title is "The Scientist and Engineer's Guide to Digital Signal Processing" and not

​

Big fat book ! But the website says " Available soon! " -- ah, there is some PDF

On p 561 he explains that addition and subtraction are OK (because real and imaginary parts are not mixed).
For multiplication things do mix up in the way he presents it.

I am very unhappy with the way he treats this material -- and clearly confuses you.
It already goes badly wrong on p 560. It is nonsense to equivalence a real function of $\omega t$ to a complex number $a+jb$It's beyond the PF scope to rewrite his book, but the way it should be done is to present the real sinusoid $$A\cos(\omega t)+ B\sin(\omega t) = M \cos(\omega t + \phi) $$ as the real part of the complex number (function of M, $\phi, \omega$) $$M e^{j\omega t + \phi} $$ (with M, the amplitude, $\phi$ and $\omega$ all real numbers)

This way complex multiplication and all other kinds of manipulation work just fine, something he happily uses on p 562.

You are right: Yes, it is. And it confuses readers no end. We should send him a link to your post as feedback -- perhaps he can improve on it ...

'Highly rated' it may be, but I'd go look for a better book / pdf ) There's plenty around.

Yes this is it. He gives each chapter as a PDF for free on the site and its got many reviews on amazon .com

He does mention there is two ways to use complex numbers and sinusoids, equivalence and substitution and he says that entire chapter will be in regard to substitution on pg 558, second paragraph.

Yes I am confused and not clear what I should be interpreting with addition and multiplication of complex sinusoids.

Does anyone recommend a text or can give guidance? I need to know learn interpretations and pitfalls of operations like multiplying complex sinusoids of different frequencies, adding complex sinusoids of different frequencies...

 

[FactChecker]

Multiplication of complex numbers produces a number whose real and imaginary parts are combinations of the real and imaginary parts of the individual numbers that were multiplied. Simple multiplication of sin functions is like only multiplying the imaginary parts of complex numbers and does not include other important terms. So those two types of multiplication are not equivalent.
That being said, a complex number is a very good way to represent r*cos() and r*sin() as the real and imaginary parts. And multiplying two complex numbers gives another complex number, representing other r*cos() and r*sin() functions. The real part is $$r_\alpha*cos(\alpha*t)*r_\beta*cos(\beta*t)-r_\alpha*sin(\alpha*t)*r_beta*sin(\beta*t) = r_\alpha*r_\beta*cos((\alpha+\beta)*t)$$ So the real part of the product is a cos() function by the sum-angle formula.
Similarly, the imaginary part of the product is a sin function by the sum-angle formula: $$r_\alpha*r_\beta*sin((\alpha+\beta)*t)$$
Simply multiplying two sin() functions or two cos() functions does not give such a nice result.

 

[Natalie Johnson]

Further to my previous post. He also says on page 561 Chapter 30 'Complex numbers'
http://www.dspguide.com/ch30.htm

Complex sinusoids must have the same frequency to be added ... which is not the case for real sinusoids? I mean I can add two sinusoids of different frequencies quite easily in MATLAB... So why not with complex?

Multiplication of complex numbers produces a number whose real and imaginary parts are combinations of the real and imaginary parts of the individual numbers that were multiplied. Simple multiplication of sin functions is like only multiplying the imaginary parts of complex numbers and does not include other important terms. So those two types of multiplication are not equivalent.
That being said, a complex number is a very good way to represent r*cos() and r*sin() as the real and imaginary parts. And multiplying two complex numbers gives another complex number, representing other r*cos() and r*sin() functions. The real part is $$r_\alpha*cos(\alpha*t)*r_\beta*cos(\beta*t)-r_\alpha*sin(\alpha*t)*r_beta*sin(\beta*t) = r_\alpha*r_\beta*cos((\alpha+\beta)*t)$$ So the real part of the product is a cos() function by the sum-angle formula.
Similarly, the imaginary part of the product is a sin function by the sum-angle formula: $$r_\alpha*r_\beta*sin((\alpha+\beta)*t)$$
Simply multiplying two sin() functions or two cos() functions does not give such a nice result.

I see the maths and I am okay with it, but the interpretation is confusing me and how it relates to the physical system once its taken out of complex form

 

[scottdave]

It sounds as if the reference about must be same frequency refers to using a complex number to represent a magnitude and phase of a signal of certain frequency. The math of manipulating these is simplified by representing them as a complex number.

 

[FactChecker]

Further to my previous post. He also says on page 561 Chapter 30 'Complex numbers'
http://www.dspguide.com/ch30.htm

Complex sinusoids must have the same frequency to be added ... which is not the case for real sinusoids? I mean I can add two sinusoids of different frequencies quite easily in MATLAB... So why not with complex?I see the maths and I am okay with it, but the interpretation is confusing me and how it relates to the physical system once its taken out of complex form

I'm afraid that I have been careless. Any non-zero complex number can be represented as $z = r*e^{i\omega}$, but that is a constant number. To keep things simple, let's set $r = 1$. When you talk about sinusoidal functions, the complex-valued function of time t: $f(t) = e^{t*i\omega_f} = cos(t*\omega_f) + i*sin(t*\omega_f)$ is a good way of representing a cos() or sin() function by the respective real and imaginary parts. Suppose we have another similar function, $g(t) = e^{t*i\omega_g} = cos(t*\omega_g) + i*sin(t*\omega_g)$. Then their complex product is $h(t) = f(t)*g(t) = e^{t*i(\omega_f+\omega_g)} = cos(t*(\omega_f+\omega_g)) + i*sin(t*(\omega_f+\omega_g))$, whose real and imaginary parts are cos() and sin() functions again. If you tried to miltiply cos() functions or sin() functions as real numbers, you would not get anything as nice.
The situation with addition is not as nice for either the real or complex functions. Adding two complex functions just adds the real parts to get the real sum and adds the complex parts to get the complex sum. It is perfectly legal to add real sinusoidal functions, but the sum will not be sinusoidal. Similarly, it is perfectly legal to add complex exponential functions, but the sum will not be exponential.

 

[Mark44]

Any non-zero complex number can be represented as $z = r*e^{i\omega}$

And zero can be represented in the same way, with r = 0.

 

[BvU]

Does anyone recommend a text or can give guidance? I need to know learn interpretations and pitfalls of operations like multiplying complex sinusoids of different frequencies, adding complex sinusoids of different frequencies...

I recall a short writeup by @LCKurtz title There’s nothing imaginary about complex numbers. It's more for teachers but there are similarities with Smith (like in Kurtz 3.1). It sure has the advantage of being al lot more concise !
Lynn's other writeup alternating current impedance is also quite good in your context.