All,
At the moment I am trying to follow the process of converting the Fourier series into the fourier transform. I am stuck on the middle bit atm, the complex fourier series. I have been trying my self but have been hitting a brick wall. I would appreciate your help.
I am taking this from the book: Fourier series by W.Bolton,
ISBN 0-582-23934-6, page 108.
A periodic rectangular pulse is considered with a period of \(\tau\), the amplitude is V. The complex Fourier series for this function is:
\(Cn=\frac{1}{2}\int f(t) e^{-jnwot} dt\) (the limits are T/2 and -T/2)
The next step includes the rectangular function which does not span the entire range of T, therefore limits have changed.
\(Cn=\frac{1}{2}\int V e^{-jnwot} dt\) (the limits are \(\tau\)/2 and -\(\tau\)/2)
\(=\frac{V}{T}[\frac{e^{-jnwot}}{-jnwo}]\) (the limits are \(\tau\)/2 and -\(\tau\)/2)
frequency of the fundamental is wo =2π/T
\(Cn = -\frac{V}{j2n\pi}(e ^{-jn\pi/T}-e ^{jn\pi/T})\)
\(Cn = \frac{V}{n\pi} sin \)\(\frac{n\pi\tau}{T}\)
I understand the steps so far but do not understand the next step. which states that Cn can be re-written as:
\(Cn = \frac{V\tau}{T}\)\(\frac{sin \frac{n\pi\tau}{T}}{\frac{n\pi\tau}{T}}\)
I dont understand how this sinc x function was derived especially when the book states that when n=0, the amplitude is \(A = \frac{V\tau}{T} \). Surely its 0.
I entered \(Cn = \frac{V}{n\pi} sin \)\(\frac{n\pi\tau}{T}\) into MATLAB and it shows no value for when n=0 as expected but the book reckons it exists. Does anyone know why?
Thanks in advance.
At the moment I am trying to follow the process of converting the Fourier series into the fourier transform. I am stuck on the middle bit atm, the complex fourier series. I have been trying my self but have been hitting a brick wall. I would appreciate your help.
I am taking this from the book: Fourier series by W.Bolton,
ISBN 0-582-23934-6, page 108.
A periodic rectangular pulse is considered with a period of \(\tau\), the amplitude is V. The complex Fourier series for this function is:
\(Cn=\frac{1}{2}\int f(t) e^{-jnwot} dt\) (the limits are T/2 and -T/2)
The next step includes the rectangular function which does not span the entire range of T, therefore limits have changed.
\(Cn=\frac{1}{2}\int V e^{-jnwot} dt\) (the limits are \(\tau\)/2 and -\(\tau\)/2)
\(=\frac{V}{T}[\frac{e^{-jnwot}}{-jnwo}]\) (the limits are \(\tau\)/2 and -\(\tau\)/2)
frequency of the fundamental is wo =2π/T
\(Cn = -\frac{V}{j2n\pi}(e ^{-jn\pi/T}-e ^{jn\pi/T})\)
\(Cn = \frac{V}{n\pi} sin \)\(\frac{n\pi\tau}{T}\)
I understand the steps so far but do not understand the next step. which states that Cn can be re-written as:
\(Cn = \frac{V\tau}{T}\)\(\frac{sin \frac{n\pi\tau}{T}}{\frac{n\pi\tau}{T}}\)
I dont understand how this sinc x function was derived especially when the book states that when n=0, the amplitude is \(A = \frac{V\tau}{T} \). Surely its 0.
I entered \(Cn = \frac{V}{n\pi} sin \)\(\frac{n\pi\tau}{T}\) into MATLAB and it shows no value for when n=0 as expected but the book reckons it exists. Does anyone know why?
Thanks in advance.