Whether plotting a spectrum of periodic signal using Fourier transform and Fourier series result the same shape? (what is continuous spectra and line spectra?).
Dear Arul, good question
The concept is that there are two descriptions of a signal according to Fourier. The time domain in form of time waveform say f(t) and the frequency domain form F(w) in form of a combination of complex periodic signal with frequency as a variable instead of t.
Physically, every time waveform can be analysed into frequency components called the frequency of the waveform.
If the time waveform is periodic the resulting frequency components will have discrete frequencies with integral multiples of the periodic frequency of the waveform, that is the fundamental and the harmonics, The nature of the waveform f(t) determines the allowed values of the frequencies of the resulting periodic exponential signals or the sine and cosine waves. It is observed that if time waveform is periodic with frequency w, the resulting frequency components will be discrete with frequencies w,2w,3w ....etc.
In case of a nonperiodic signal as the case of the signal pulse the the Fourier transform
will contain all possible frequencies continuously extending from mince infinity to plus infinity.
The two signal representations are equivalent and one is the transform or the inverse transform of the other.
So, conversely one can synthesize the periodic f(t) by summing up a series of sine and cosine waves with frequency in harmonic relationship.
While a single nonperiodic pulse can be synthesized by frequency components with frequency covering all possible values from mince infinity to plus infinity continuously.
In summary, a transform is mathematical operation that is when applied results in two equivalent description of the function.
It is so that the signal are generated in time domain and transformed in frequency domain to analyse them and make them more perceivable.After performing the transform operation one can interpret the resulting image no more, no less. This is because the transform is unique, which means that there is one to one correspondence between the function and its image.
The Fourier transform is a very powerful tool in analyzing and synthesizing the signals. The concept of the bandwith is the consequence of the Fourier transform.
as every signal in time domain has its frequency content or spectrum.
In fact, from the point of view of signal synthesis according to Fourier, the sinewave can be considered the primitive function, from which any function can be built just by combining.
In conclusion, we get the image of the transform according the mathematical formulation and then we interpret the result.
Best wishes
Dear Arul, good question
The concept is that there are two descriptions of a signal according to Fourier. The time domain in form of time waveform say f(t) and the frequency domain form F(w) in form of a combination of complex periodic signal with frequency as a variable instead of t.
Physically, every time waveform can be analysed into frequency components called the frequency of the waveform.
If the time waveform is periodic the resulting frequency components will have discrete frequencies with integral multiples of the periodic frequency of the waveform, that is the fundamental and the harmonics, The nature of the waveform f(t) determines the allowed values of the frequencies of the resulting periodic exponential signals or the sine and cosine waves. It is observed that if time waveform is periodic with frequency w, the resulting frequency components will be discrete with frequencies w,2w,3w ....etc.
In case of a nonperiodic signal as the case of the signal pulse the the Fourier transform
will contain all possible frequencies continuously extending from mince infinity to plus infinity.
The two signal representations are equivalent and one is the transform or the inverse transform of the other.
So, conversely one can synthesize the periodic f(t) by summing up a series of sine and cosine waves with frequency in harmonic relationship.
While a single nonperiodic pulse can be synthesized by frequency components with frequency covering all possible values from mince infinity to plus infinity continuously.
In summary, a transform is mathematical operation that is when applied results in two equivalent description of the function.
It is so that the signal are generated in time domain and transformed in frequency domain to analyse them and make them more perceivable.After performing the transform operation one can interpret the resulting image no more, no less. This is because the transform is unique, which means that there is one to one correspondence between the function and its image.
The Fourier transform is a very powerful tool in analyzing and synthesizing the signals. The concept of the bandwith is the consequence of the Fourier transform.
as every signal in time domain has its frequency content or spectrum.
In fact, from the point of view of signal synthesis according to Fourier, the sinewave can be considered the primitive function, from which any function can be built just by combining.
In conclusion, we get the image of the transform according the mathematical formulation and then we interpret the result.
Best wishes
Dear Arul,
Fourier series gives the frequency content of a periodic continuous-time signal such as f(t) with a period T. Inverse of the period (1/T) is called the fundamental frequency of f(t). In Fourier domain, frequency components of f(t) are placed at integer multiples of the fundamental frequency; namely at k/T where k is an integer between -infinity and +infinity. As a result, spectrum of a periodic signal which is obtained via Fourier series is discrete in natüre. Notice that these components are separated in frequency by 1/T.
On the other side of the coin, the Fourier transform gives frequency content of an aperiodic continuous-time signal such as g(t). Since aperiodic, in the limit, g(t) can be considered as periodic with a period of infinity. Thus, the separation of its frequency components in Fourier domain goes in the limit to (1/infinity)=0. Hence, the spectrum of an aperiodc signal has a continuous spectrum.
In summary, in Fourier series, the time domain signal is continuous and periodic and its Fourier spectrum is discrete and aperiodic.
In Fourier transform, the time domain signal is continuous and aperiodic and its Fourier spectrum is also continuous and and aperiodic.
Best wishes,
Fourier series gives the frequency content of a periodic continuous-time signal such as f(t) with a period T while Fourier transform gives frequency content of an aperiodic continuous-time signal such as X(t). Since aperiodic signal X(t) can be expressed as periodic with a period of infinity, the separation of its frequency components in Fourier domain goes in the limit to (1/infinity)=0. Therefore, the power spectrum of an aperiodc signal will always have a continuous spectrum.
I would like to thank all colleagues who upvoted my answer to this very interesting question. From my point of view, the Fourier transform is a fundamental tool in signal analysis and signal synthesis. If the description of an object is abstract like the time waveform, one can perceive little from it. If one can analyse the object, one would know much more about it. This is the great issue with the Fourier transform. in the frequency domain, the signal in an analysed form, which means that it is resolved to its frequency components which are the building units of the waveform.
The concept of the bandwidth in communications is a straight forward result of the frequency domain description of the signals.
Many signal processing can be done easier in frequency domain. The measurements of the performance is also easier in frequency domain. Time domain and frequency domain are dual.
Thank you again
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