- Trigonometric Fourier Series Definition: The trigonometric Fourier series is defined as a method to represent periodic signals using sine and cosine functions, derived from the exponential form.
- Fourier Coefficients: The coefficients ak and bk determine the contribution of each sine and cosine term in the series.
- DC Component: The term a0 is the DC component, representing the average value of the signal.
- Even and Odd Functions: For even functions, bk = 0, and for odd functions, a0 = 0 and ak = 0.
- Effect of Signal Shifting: Shifting the signal horizontally changes the phase spectrum, while vertical shifts change the DC value.
The Fourier series can be written in complex exponential form or as real sine and cosine terms. This page explains the Trigonometric Fourier series for periodic signals.
Fourier series representation in Trigonometric form
Fourier Series in Trigonometric Form: Euler’s identities combine positive- and negative-frequency complex exponentials into real cosine and sine terms. For a periodic signal x(t) with fundamental period To, the complex exponential series is shown below.
For a real signal, conjugate symmetry of the complex coefficients produces real trigonometric coefficients. Convergence and the value at a discontinuity depend on the signal’s regularity conditions.
The trigonometric Fourier series represents a periodic signal x(t) as a constant term plus harmonics at integer multiples of the fundamental frequency.
The coefficients ak and bk give the cosine and sine contributions for harmonic k.
The constant a0 represents the DC or average component under the convention used in the displayed series.
Properties of Fourier series
1. If x(t) is an even function, x(-t) = x(t), then every sine coefficient vanishes: bk = 0 and
2. If x(t) is an odd function, x(-t) = -x(t), then the DC and cosine coefficients vanish: a0 = 0, ak = 0 and
3. If x(t) has half-wave symmetry, x(t) = -x(t ± T0/2), then its average and even harmonics vanish: a0 = 0, ak = bk = 0 for k even,
4. Linearity – A linear combination of signals produces the same linear combination of their coefficients.
5. Time shifting – A time shift preserves each harmonic magnitude and changes its phase by a frequency-dependent amount.
6. Time reversal – Replacing t by -t preserves cosine terms and reverses the sign of sine terms.
7. Multiplication – Pointwise multiplication in time corresponds to a convolution of series coefficients under the stated normalization.
8. Conjugation – Complex conjugation produces the corresponding conjugate and index-reversal relationship between coefficients.
9. Differentiation – Where term-by-term differentiation is valid, the kth harmonic coefficient is multiplied by its angular-frequency factor.
10. Integration – Periodic integration divides each nonzero harmonic by its angular-frequency factor; the DC term needs separate treatment.
11. Periodic Convolution – Periodic convolution corresponds to multiplication of matching series coefficients, with a scale factor set by the convention.
Relationship between coefficients of exponential form and coefficients of trigonometric form
For real x(t), the trigonometric coefficients are real and the complex exponential coefficients have conjugate symmetry.
Effect of Shifting Axis of the Signal
- Shifting a periodic waveform left or right in time preserves the magnitude of every harmonic. It changes the phase of harmonic k by an amount proportional to k and the time shift.
- Adding or subtracting a constant amplitude changes only the DC coefficient. The nonzero harmonic coefficients remain unchanged.
