Takaitaccen Exponential Fourier Series

Fourier Series at a Glance

Zaɓi mai yaushe a lokacin daɗi x(t) ce mai yaushe mai yawa idan akwai ƙarin ɗaya da ta fi T tana cika daga ita

Idan an san ne ake kula da ma'adanci mai yaushe a kan sinusoide ko eksponenshiko mai yawa, idan ya ci gaba da Dirichlet’s Conditions. Wannan bayanin da ake kula shi ana kiran da FOURIER SERIES.
Biyu naɗa Fourier Series da suka da su. Duk da cewa biyu suna da mu'amala da waɗanda suka da su.

• Eksponenshiko Fourier Series

• Trigonometriko Fourier Series

Duka biyu naɗa suna da abubuwa mafi yawan da su. Idan an samun wani abu, za a zabi wani naɗa da aka da su saboda dalilin da aka da su.

Mai yaushe mai yawa ana kula shi a kan Eksponenshiko Fourier Series a kan waɗannan uku naɗa:

1. Bayanin Mai Yaushe Mai Yawa.

2. Amplitudin da Phase Spectra na Mai Yaushe Mai Yawa.

3. Power Content na Mai Yaushe Mai Yawa.

Bayanin Mai Yaushe Mai Yawa

Mai yaushe mai yawa a kan Fourier Series zai iya bayarwa a biyu naɗa tsarin lokaci:

1. Tsarin Lokaci Mai Tsawo.

2. Tsarin Lokaci Mai Yau.

Tsarin Lokaci Mai Tsawo

Akwai Eksponenshiko Fourier Series bayanin mai yaushe mai yawa x(t) da tunanen lokaci mai yawa To ya kasance da

Idan C ya kasance da Complex Fourier Coefficient da ya kasance da,

Idan ∫0T0, na nufin integral over any one period and, 0 to T0 ko –T0/2 to T0/2 su na iya amfani da su don integral.
An samun equation (3) tare da mutumin sarautar zuwa equation (2) da e(-jlω0t) kuma integral over a time period both sides.

Idan an karin order of summation and integration on R.H.S., muna samu



Idan k≠l, right hand side of (5) evaluated at the lower and upper limit yields zero. On the other hand, if k=l, we have

Consequently equation (4) reduces to



which indicates average value of x(t) over a period.
When x (t) is real,

Where, * indicates conjugate

Tsarin Lokaci Mai Yau

Fourier representation in discrete is very much similar to Fourier representation of periodic signal of continuous time domain.
The discrete Fourier series representation of a periodic sequence x[n] with fundamental period No is given by
Where, Ck, are the Fourier coefficients and are given by

This can be derived in the same way as we derived it in continuous time domain.

Amplitude and Phase Spectra of a Periodic Signal

We can express Complex Fourier Coefficient, Ck as

A plot of |Ck| versus the angular frequency w is called the amplitude spectrum of the periodic signal x(t), and a plot of Фk, versus w is called the phase spectrum of x(t). Since the index k assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies kω0, they are therefore referred to as discrete frequency spectra or line spectra.
For a real periodic signal x (t) we have C-k = Ck*. Thus,