fourier and laplace examples

[PG1995]

Hi

Could you please help me with these queries, Q1, Q2, Q3, and **broken link removed**. If you can't view the images, then please use these details : username: imgshack4every1 and password: imgshack4every1. Thank you.

Regards
PG

 

[steveB]

For Q1, you did not substitute in for u correctly in the last line.

For Q2, those types of integrals can be solved. I normally use integral tables to save time on these. Even if you want to work it out yourself, it is good to check an integral table just to make sure the integral is solvable, and be sure you are not wasting your time on an unsolvable integral. Also, symbolic calculations programs can help do the same. Often, I use all three methods. I check a table. Then I use a symbolic processor to make sure the table did not have a typo. I have found typos in tables before and I've also found symbolic processors that give the wrong answer. Then I work it out myself at my leisure. You can also check again with a numerical integration, just to be sure.

 

[PG1995]

Thanks a lot.

Q1: I did make a mistake in writing out the upper limit at the end but even correcting it doesn't make the end solution correct. I'm sorry but I couldn't trace any other error.

Q2: Yes, my formulas book has a listing for such integrals but I was asking it generally. I'm sure it needs some effort to do that integral(s) from scratch.

**broken link removed**

I have just noticed that a key word is missing in Q4: "But I don't know how to convert the final solution...". Thank you.

Regards
PG

 

[steveB]

Q1: I did make a mistake in writing out the upper limit at the end but even correcting it doesn't make the end solution correct. I'm sorry but I couldn't trace any other error.

It looks to me that yours and the book's solution are now both doing the same thing. However, the book solution appears to have a typo in the last step.

Q2: Yes, my formulas book has a listing for such integrals but I was asking it generally. I'm sure it needs some effort to do that integral(s) from scratch.

**broken link removed**

The effort required depends on the approach taken. However, I would think that converting sine or cosine to complex exponential form using the Euler formula leads to a fairly easy solution. There are a number of steps with this approach, since it is not an elegant or inspired method, but if you carefully write it out, you should be able to derive the formulas you find in the integral table.

You can also do this one fairly easily using integration by parts. [latex]\int u\;dv=uv-\int v\;du [/latex]

 

[PG1995]

Thanks a lot.

When you have time please also try to help me with the Q3 and **broken link removed**. I think I should mention it again that a key word is missing in Q4: "But I don't know how to convert the final solution...". Thank you.

Regards
PG

 

[steveB]

Thanks a lot.

When you have time please also try to help me with the Q3 and **broken link removed**. I think I should mention it again that a key word is missing in Q4: "But I don't know how to convert the final solution...". Thank you.

Regards
PG

For Q3, the answer looks correct. You can write the coefficients in a simpler way. bn=2/(n∏) for odd n, bn=0 for even n

 

[steveB]

Thanks a lot.

When you have time please also try to help me with the Q3 and **broken link removed**. I think I should mention it again that a key word is missing in Q4: "But I don't know how to convert the final solution...". Thank you.

Regards
PG

For Q4, you have the ability to check the answer yourself. You show the solution for the "a" and "b" coefficients and you show the formula for "c" when you have the "a" and "b" coefficients. Just check it with algebra.

I can't help more on this one because I'm having trouble reading it.

 

[PG1995]

Hi

I don't have my computer with me right now and it will take two or three days to get it repaired because it just crashed yesterday! Using my phone to make this post.

Could you please tell how I can transfer the end solution into proper Fourier series consisting of sine and cosine terms? In **broken link removed** you can see it shows the conversion between different forms but I don't get it how I transfer exponential form into sine-cosine form.Thank you.

My end solution for **broken link removed** should be 2sin(bw)/w. Where did I go wrong? Please help me.

Regards
PG

 

[steveB]

Could you please tell how I can transfer the end solution into proper Fourier series consisting of sine and cosine terms? In **broken link removed** you can see it shows the conversion between different forms but I get it how I transfer exponential form into sine-cosine form.

This is an example where understanding is paramount. If you truly understood the meaning of the formulas, you would not need to ask this. Blind application of formulas is not that useful if you don't know how to interpret what you are doing and what the result means.

The exponential form is closely related to the cosine with angle form. The Cn and θn are the magnitude and phase portions of the complex number you get using the complex exponential form. The An and Bn numbers are closely related to the sin/cos form because these are like rectangular components and the Cn and θn are like polar form coordinates. The transformation is the usual magnitude (square root of the sum of the squared compnonents) and phase (atan function).

A key part of engineering is learning how to develop methods of checking on your own. You can't keep asking others to check your answers because eventually there will be no one who knows the answer to your problem. If they need to go back and check if you are correct, well they might as well have done it themselves. However, I understand that in the learning phases you sometimes need some guidance. Here, my guidance is to look at this carefully and figure out the conversions yourself. This is the only way to know that you know the fundamentals and the meaning of the formulas.

In the second part of your question I like that you know your form does not match, and you are trying to figure out why they don't match. I'll look at this problem soon.

 

[PG1995]

Hi Steve

I understand your advice and you have a good point. I always try to do my best to understand things with in the limited time and using my limited knowledge. For this particular case I don't think I'm applying the formula totally blindly. I'm sure the end solution can be converted into sine and cosine form but I'm equally sure that the conversion is not that much straightforward. You might find it easy conversion but for a person like me it won't be easy. I don't have my computer with me otherwise I have showed it to you where I'm having problem. The problem I'm facing is more about mathematical manipulation one rather than conceptual one.

I'm sorry to ask this but I'm also **broken link removed** with Fourier transform of impulse function. If possible, please let me know where I'm going wrong. Thank you.

Best wishes
PG

 

[steveB]

My end solution for **broken link removed** should be 2sin(bw)/w. Where did I go wrong? Please help me.

I see two issues. First, your limits of integration should be -b to +b and not 0 to b.

Second, the Fourier transform (i) is correct, but the versions (ii) and (iii) appear to me to be the Fourier Sine Transform and the Fourier Cosine Transform, which are not quite the same thing. Formulas (ii) and (iii) are certainly related and can be used to construct the full transform, but I believe you have misinterpreted their meaning.

 

[steveB]

The problem I'm facing is more about mathematical manipulation one rather than conceptual one.

I suspect that this is not true. The mathematics is just the basic math of polar to rectangular coordinate conversion, which I believe you are familiar with. The problem may be conceptual in that you do not "see" the direct connection of polar/rectagular coordinates and how this relates to complex numbers. The only way to break through the conceptual barrier is to work it all out on paper and in your mind.

At the very least you should be able to work out the conversion formulas. After that, if you have trouble doing the math on the particular problem you are trying to check for consistancy, I can try to help with that.

I'm sorry to ask this but I'm also **broken link removed** with Fourier transform of impulse function. If possible, please let me know where I'm going wrong. Thank you.

You are going wrong in your definition of the Dirac delta function. This gets back to conceptual problems again. We have been discussing the Dirac delta function with you extensively and we have noted that it is a very strange limiting function with the property that the integral of it is one. But, now you go and write that δ(t)=1 at t=0. How can you make this blunder? It seems the concepts are not sticking in your mind.

Please start with the correct definition of the delta function. Without that starting point, you have no hope of getting the correct answer.

 

[PG1995]

My sincerest apologies about that delta function blunder. I don't know why my mind was completely out of focus.

As for the other issue of conversion, I think I should postpone the discussion until I get my computer back.

In that problem of Fourier transform of rect function I knew that I was supposed to use limits from -b to +b but that **broken link removed** didn't allow it. Perhaps, those sine and cosine formulas for Fourier transform are not to be used here, and the exponential formula should only be used. Please note that in the linked scan from a formula book there is no cosine formula for Fourier transform because it was given on the next page. Yes, here I confess that I'm using these formulas somewhat blindly. But personally I don't entirely blame myself. We are now using no book for the signal and system course. The instructor uses random slides from the net. So, even when my end answer for that rect function came to be wrong I knew that I have reached correct solution had I used the exponential form but I was curious to know why I can't do it with the sine formula.

Thank you for your patience and help.

Regards
PG

 

[steveB]

My sincerest apologies about that delta function blunder. I don't know why my mind was completely out of focus.

No need to apologize. We all do this from time to time. It's important for a coach, teacher, tutor, or (in this case) friend trying to help, to point out when the focus is being lost. We do this, not to be unkind, but to help you focus better.

In that problem of Fourier transform of rect function I knew that I was supposed to use limits from -b to +b but that **broken link removed** didn't allow it. Perhaps, those sine and cosine formulas for Fourier transform are not to be used here, and the exponential formula should only be used. Please note that in the linked scan from a formula book there is no cosine formula for Fourier transform because it was given on the next page. Yes, here I confess that I'm using these formulas somewhat blindly. But personally I don't entirely blame myself. We are now using no book for the signal and system course. The instructor uses random slides from the net. So, even when my end answer for that rect function came to be wrong I knew that I have reached correct solution had I used the exponential form but I was curious to know why I can't do it with the sine formula.

I understand the difficulties inherent in this situation. I recommend the following book which is relatively inexpensive in paperback or electronic form, but has good information and examples.

https://www.amazon.com/Schaums-Outl...95&sr=8-5&keywords=Signals+and+systems+Schaum

The above reference I know to be good. I noticed the following reference which might help you. However, I don't have this book, so I can't be sure how good it is.

https://www.amazon.com/Signals-Systems-Demystified-ebook/dp/B008MAO2A4/ref=pd_sim_kstore_1

 

[PG1995]

Thank you for the recommendations, Steve.

I have used both of those books and have them in electronic form. The problem is that in the class we don't use any particular book; the instructor just uses random slides. In my view this is not a good way of teaching.

Q1:
Anyway, I have been through the problem from this post which prompted all this discussion that day. Please have a look here.

Q2:
Could you please help me with this query? Thanks very much.

Regards
PG

 

[steveB]

For Q2, the sign error occurs when you take the anti-derivative of the exponential. You need a negative sign at that point.

 

[steveB]

For Q1, I think I've lost track of what your exact question is. Are you just trying to verify the answer you got? Or, are you trying to make sure you know how to convert between coefficients?

If trying to verify the answer, a quick way is to write a short Matlab program that calculates and compares the different formulas for the coefficients using the conversion formulas. In this way you don't have to try to manipulate formulas and make sure they are mathematically equivalent.

Generally, it's good if you are able to mathematically prove equivalence, but there is no certainty that this will always be easy or fast to do. Numerical checking is a shortcut that is acceptable for an engineer to double check answers.

Anyway, if you can clarify the question specifically, I can try to address it better. However, overall it seems that you are spending too much time on a relatively simple problem. But again, maybe you want to be sure that you have the correct foundations in place, which is always a good use of time.

 

[PG1995]

Thank you very much.

In this post I asked that how I can transform the end solution of the exponential Fourier series into sine-cosine Fourier series. Then, you thought that I was applying the formulas blindly and in your view conversion from the exponential into sinusoid Fourier form was easy. Then I went through it again and found that the conversion was not easy, at least for me. I was saying that the problem I was facing was more of a math manipulation one rather than a conceptual one. Now here (the expression in light green highlight at the start of second to last line is the end solution) I'm trying to prove my point again that the conversion is not easy. I hope it's clear now. If not, then please let me know. Perhaps, I could put it better then. Thanks a lot.

Regards
PG

 

[steveB]

Thank you very much.

In this post I asked that how I can transform the end solution of the exponential Fourier series into sine-cosine Fourier series. Then, you thought that I was applying the formulas blindly and in your view conversion from the exponential into sinusoid Fourier form was easy. Then I went through it again and found that the conversion was not easy, at least for me. I was saying that the problem I was facing was more of a math manipulation one rather than a conceptual one. Now here (the expression in light green highlight at the start of second to last line is the end solution) I'm trying to prove my point again that the conversion is not easy. I hope it's clear now. If not, then please let me know. Perhaps, I could put it better then. Thanks a lot.

Regards
PG

OK, I'm not sure if the mathematical proof/manipulation is easy or hard. I'm going to try it now and see for myself. I'll post my opinion soon.

However, you can just apply the formulas and not worry about how elegant the final form looks.

Also, the fastest way to just check the answer is check it numerically in Matlab.

Just taking a quick look, I think you may be off by a factor of 2. However, I'll be more sure about this soon.

 

[steveB]

OK, so I think the math manipulations needed here are just standard techniques needed when dealing with complex math problems.

I took the Cn definition and determined the following formula.

[latex] c_k=\frac{1}{\pi}\cdot \int_0^\pi e^{-t\cdot(0.5+j\cdot 2\cdot k)}\cdot dt=\frac{2\cdot (1-e^{-\pi /2})}{\pi\cdot (1+j\cdot 4\cdot k)},\ \ {\rm for \ all} \ k[/latex]

The standard trick when you have a complex number in the denominator is to multiply both the numerator and the denominator by the complex conjugate of the denominator. This leads to the following form.

[latex] c_k=\frac{2\cdot (1-e^{-\pi /2})\cdot (1-j\cdot 4 \cdot k)}{\pi\cdot (1+16\cdot k^2)} \approx \frac{0.504\cdot (1-j\cdot 4 \cdot k)}{(1+16\cdot k^2)},\ \ {\rm for\ all} \ k[/latex]

From here it is easy to see the form of the "a" and "b" coefficients for the rectangular cosine/sine form, because the conversions are something like the following.

[latex] c_k=\frac{1}{2}\cdot (a_k-j\cdot b_k),\ \ {\rm for} \ \ k=1,2,3 ...[/latex]
[latex] c_{-k}=\frac{1}{2}\cdot (a_k+j\cdot b_k),\ \ {\rm for} \ \ k=1,2,3 ...[/latex]

and the answers given before were as follows.

[latex] a_k=0.504\cdot \frac{2}{1+16\cdot k^2}[/latex]
[latex] b_k=0.504\cdot \frac{8\cdot k}{1+16\cdot k^2}[/latex]

The zero coefficient can also be handled directly, but I've seen two forms of Fourier series with either ao or ao/2. As long as you are consistent with definitions, conversions and usage, the alternate forms will work out correctly.