Hi I am trying to learn/get to grips with Fourier Analysis by analysing a simple unipolar square wave of amplitude A (=1v) using both Fourier Series and Fourier Transforms. I have used a 50% and 25% duty cycle to highlight the differences.
I know the Fourier Series (FS) are to be used with waveforms that are periodic and the series really breaks down periodic signals/waveforms into a number of time based sines and cosines. If we add enough of these sines and cosines together we should end up with a waveform very similar to the original periodic waveform. The sines and cosines all have different amplitudes and occur at multiples of the fundamental frequency. We can calculate the average power of contributed by each of the sines/cosines thereby giving a discrete amplitude spectra of the periodic waveform, in watts, dB or dBm.
The Fourier Transform (FT) is to be used on aperiodic signals and produces a frequency dependent formula which will return a value in the units of volts seconds if an input frequency is applied. The frequency dependent formula thereby produces a continuous amplitude spectra with amplitude readings in the unusual units of volts seconds.
I am aware that if the period of a periodic waveform is increased i.e. duty cycle changes from 50% to 25% to 12.5% etc. the discrete amplitude spectra move closer together until they become a continuous waveform (Fourier Trnasform) at period = infinity
I was wondering was there any way of reconciling the amplitude spectra from the FS and FT before the period = infinity ?
The attached spreadsheet pdf’s gives some calculations and data I have used in order to reconcile the differences.
You will see the first 2 pdf’s “squarespectrum11&12” really have calculated the discrete amplitude spectra for a 50% and 25% duty cycle unipolar square wave of amplitude A (=1v) in terms of both watts and dBm for both a single line spectra (positive frequencies only) and double line spectra (positive and negative frequencies). The third pdf “squarespectrum13” plots the results for positive frequencies (single line spectra).
The fifth pdf ”squarespectrum15”, calculates the results in terms of watts and dBm using the Fourier Transform equation I developed for 50% and 25% duty cycle waveforms. You will notice these equations at the top of the page. The columns entitled “Fourier Series dBm 2 line spectra” have been imported from the first 2 pdf’s to compare the results.
The interesting thing is that for 50% duty cycle the results are exactly the same !!!
However for the 25% duty cycle the results are different. Although there are some similarities i.e. the dc component is the same and the FT spectra are 3 dBm (half power) down on most of the FS spectra ?
I am not sure if my calculations are correct ? If they are, how can the differences be explained ?
I thought it might have something to do with the fact that the FS readings are all average power, but I think the FT readings are peak values and possibly need to be divided by root 2.
PLEASE HELP
I know the Fourier Series (FS) are to be used with waveforms that are periodic and the series really breaks down periodic signals/waveforms into a number of time based sines and cosines. If we add enough of these sines and cosines together we should end up with a waveform very similar to the original periodic waveform. The sines and cosines all have different amplitudes and occur at multiples of the fundamental frequency. We can calculate the average power of contributed by each of the sines/cosines thereby giving a discrete amplitude spectra of the periodic waveform, in watts, dB or dBm.
The Fourier Transform (FT) is to be used on aperiodic signals and produces a frequency dependent formula which will return a value in the units of volts seconds if an input frequency is applied. The frequency dependent formula thereby produces a continuous amplitude spectra with amplitude readings in the unusual units of volts seconds.
I am aware that if the period of a periodic waveform is increased i.e. duty cycle changes from 50% to 25% to 12.5% etc. the discrete amplitude spectra move closer together until they become a continuous waveform (Fourier Trnasform) at period = infinity
I was wondering was there any way of reconciling the amplitude spectra from the FS and FT before the period = infinity ?
The attached spreadsheet pdf’s gives some calculations and data I have used in order to reconcile the differences.
You will see the first 2 pdf’s “squarespectrum11&12” really have calculated the discrete amplitude spectra for a 50% and 25% duty cycle unipolar square wave of amplitude A (=1v) in terms of both watts and dBm for both a single line spectra (positive frequencies only) and double line spectra (positive and negative frequencies). The third pdf “squarespectrum13” plots the results for positive frequencies (single line spectra).
The fifth pdf ”squarespectrum15”, calculates the results in terms of watts and dBm using the Fourier Transform equation I developed for 50% and 25% duty cycle waveforms. You will notice these equations at the top of the page. The columns entitled “Fourier Series dBm 2 line spectra” have been imported from the first 2 pdf’s to compare the results.
The interesting thing is that for 50% duty cycle the results are exactly the same !!!
However for the 25% duty cycle the results are different. Although there are some similarities i.e. the dc component is the same and the FT spectra are 3 dBm (half power) down on most of the FS spectra ?
I am not sure if my calculations are correct ? If they are, how can the differences be explained ?
I thought it might have something to do with the fact that the FS readings are all average power, but I think the FT readings are peak values and possibly need to be divided by root 2.
PLEASE HELP
