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Greg has kindly allowed me to post these equations which I compiled many years ago. Somehow I like them better than anything I've ever run across so maybe someone else will find them useful also.
Actually, I have given some thought to the Fourier series and how they tie in with sampled-data analysis. I believe the series is the proper way to do this. I have a textbook (Stanford prof. W W Harman, Principles of the Statistical Theory of Communication , a believe-you-me not to be scoffed at text, in fact a brilliant one, yet he starts with analyzing a signal x(t) of finite duration T with the footnote that "To make this argument strictly valid, the duration T must be allowed to approach infinity". Without further explanation. He then proceeds to use Fourier transform and time-domain math to synthesize x(t) from the sampled data, which may or may not be exact depending on the existence or non-existence of low-frequency (not high!) components.
But if we think Fourier series we find that we get an exact representation of the signal provided it can be extended to a periodic function of infinite duration, i.e x(t) = x(t+nT) for all integer n.
Note that this precludes for example the existence of a harmonic below ## 2 \pi/T ## since periodicity is then impossible. As an example, if ## x(t) = sin(\omega_1 t) + sin(1.5\omega_1 t) ## then T cannot be the inverse of the lowest harmonic which would be ## 2\pi/\omega_1) ## but must in fact be multiples of ## 6\pi/\omega_1 ##).
The Fourier series thus gives the exact x(t) in terms of the spectrum of the extended wave including the original x(t), 0<t<T. And of course the Fourier coefficients also exactly equal the complex DFT coefficients divided by T.
Well, that's kind of what got me to rewrite those old and decomposing Fourier equations in LaTex. Certainly nothing new or Earth'shattering, just wanted them typed up neatly. Any typos please inform.
## f(t) = \sum_{-\infty}^{\infty} A_n e^{j n \omega_1 t } #### A_n = (1/2 \pi) \int_0^{2 \pi} e^{j n\omega_1 t} d( \omega_1 t) ##We can also say
## f(t) = \sum_{-\infty}^\infty c_n e^{(j \omega_1 n t + \phi_n)} ##
## c_n = |A_n|. c_n ## are real and positive.
## \phi_n = angle ##
## = \tan^{-1} (Im A_n/Re A_n) #### A_{-n} = A_n^* ## always true.Therefore, ## c_{-n} = c_n ##
## \phi_{-n} = - \phi_n ##Also,
## f(t) = c_0 + 2 \sum_{n=1}^\infty c_n cos(n \omega_1 t + \phi_n ) ##
(NB the cn's are half as big as Skilling's on pp431-432)
Further,
For odd functions [f(-x) = -f(x)]
## A_n ## are purely imaginary
## \phi_n = +/-\pi/2 ##For even functions [f(-x) = f(x)]
## A_n ## are purely real
## \phi_n ## = 0 or ## \pi ##For half-wave symmetry [ f(x) = -f(x+T)]
## c_n = A_n = 0 ## for even nNB1: can the sign ambiguity in ##\phi_n ## above be removed a priori?
NB2: ## 2 c_n ## = peak value of each harmonic n
Actually, I have given some thought to the Fourier series and how they tie in with sampled-data analysis. I believe the series is the proper way to do this. I have a textbook (Stanford prof. W W Harman, Principles of the Statistical Theory of Communication , a believe-you-me not to be scoffed at text, in fact a brilliant one, yet he starts with analyzing a signal x(t) of finite duration T with the footnote that "To make this argument strictly valid, the duration T must be allowed to approach infinity". Without further explanation. He then proceeds to use Fourier transform and time-domain math to synthesize x(t) from the sampled data, which may or may not be exact depending on the existence or non-existence of low-frequency (not high!) components.
But if we think Fourier series we find that we get an exact representation of the signal provided it can be extended to a periodic function of infinite duration, i.e x(t) = x(t+nT) for all integer n.
Note that this precludes for example the existence of a harmonic below ## 2 \pi/T ## since periodicity is then impossible. As an example, if ## x(t) = sin(\omega_1 t) + sin(1.5\omega_1 t) ## then T cannot be the inverse of the lowest harmonic which would be ## 2\pi/\omega_1) ## but must in fact be multiples of ## 6\pi/\omega_1 ##).
The Fourier series thus gives the exact x(t) in terms of the spectrum of the extended wave including the original x(t), 0<t<T. And of course the Fourier coefficients also exactly equal the complex DFT coefficients divided by T.
Well, that's kind of what got me to rewrite those old and decomposing Fourier equations in LaTex. Certainly nothing new or Earth'shattering, just wanted them typed up neatly. Any typos please inform.
## f(t) = \sum_{-\infty}^{\infty} A_n e^{j n \omega_1 t } #### A_n = (1/2 \pi) \int_0^{2 \pi} e^{j n\omega_1 t} d( \omega_1 t) ##We can also say
## f(t) = \sum_{-\infty}^\infty c_n e^{(j \omega_1 n t + \phi_n)} ##
## c_n = |A_n|. c_n ## are real and positive.
## \phi_n = angle ##
## = \tan^{-1} (Im A_n/Re A_n) #### A_{-n} = A_n^* ## always true.Therefore, ## c_{-n} = c_n ##
## \phi_{-n} = - \phi_n ##Also,
## f(t) = c_0 + 2 \sum_{n=1}^\infty c_n cos(n \omega_1 t + \phi_n ) ##
(NB the cn's are half as big as Skilling's on pp431-432)
Further,
For odd functions [f(-x) = -f(x)]
## A_n ## are purely imaginary
## \phi_n = +/-\pi/2 ##For even functions [f(-x) = f(x)]
## A_n ## are purely real
## \phi_n ## = 0 or ## \pi ##For half-wave symmetry [ f(x) = -f(x+T)]
## c_n = A_n = 0 ## for even nNB1: can the sign ambiguity in ##\phi_n ## above be removed a priori?
NB2: ## 2 c_n ## = peak value of each harmonic n
