Lect5-FourierSeries.pdf
- 1. Fourier Series Presented by Dr. Amany AbdElSamea 1
- 2. Outline • Frequency Domain • Time Domain vs. Frequency Domain • Fourier Series 2
- 3. Frequency Domain • Time domain signal tells us how the real-world signal varies with time, whereas a frequency domain signal indicates the rate of change in signal values and its spectral composition • The frequency domain refers to the analysis of mathematical functions or signals with respect to frequency rather than time. • The “Spectrum” of frequency components is the frequency-domain representation of the signal. • A Spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain but time domain signals are visualized using oscilloscope. • The frequency domain is better for determining the harmonic content of a signal. • A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transform.
- 4. Time Domain vs. Frequency Domain In time domain it is difficult to figure out signal components But in frequency domain it is easy to figure out signal components especially if the signal contains narrow band components
- 5. Time Domain vs. Frequency Domain In time domain the noise frequency is added to the original signal But in frequency domain it is easy to differentiate between signal and noise so signal to noise characteristics is improved when interpreting the signal
- 6. Frequency Spectrum • Distribution of the amplitudes and phases of each frequency component against frequency • Frequency domain analysis is mostly used to signals or functions that are periodic over time
- 7. Frequency Transformations • The process of obtaining frequency domain characteristic equation is known as transformation. Fourier Series : It is used for analysis of periodic signals Fourier Transform: It is used for analysis of non-periodic as well as periodic signals Laplace Transform: It is used for design purpose Z transform: it is used for design purpose but for discrete time systems
- 8. Fourier Theorem Joseph Fourier 1768 to 1830
- 10. Dirichlet Conditions Any periodic signal can be classified into harmonically related sinusoids or complex exponential, provided it satisfies the Dirichlet’s conditions which are: 1- Signal should have finite number of maxima and minima over the range of time period 2- Signal should have finite number of discontinuities over the range of time period 3- Signal should be absolutely integrable over the range of time period One maxima and one minima Infinite maxima and infinite minima so FS will not exist
- 11. Types of Fourier Series • Trigonometric Fourier Series • Complex Exponential Fourier Series • Cosine with phase Fourier Series
- 13. Examples ω
- 15. Example cont.,
- 16. Example cont., Sometimes a0, an, bn may be equal to zero according to the type of signal • When signal x(t) is symmetric about t axis so a0=0 • When x(t) is even signal, there will be no sine term as sine is an odd signal and this means bn =0 • When x(t) is odd signal, there will be no cosine term as cosine is an even signal and this means that an=0.
- 17. Questions