Chapter 2

1. Chapter 2 Orthogonal Representation, Fourier Series and Power Spectra • Orthogonal Series Representation of Signals and Noise • Orthogonal Functions • Orthogonal Series • Fourier Series. • Complex Fourier Series • Quadrature Fourier Series • Polar Fourier Series • Line Spectra for Periodic Waveforms • Power Spectral Density for Periodic Waveforms Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

2. where Orthogonal Functions • Definition:Functions ϕn(t) and ϕm(t) are said to be Orthogonal with respect to each other the interval a < t < b if they satisfy the condition, • δnm is called the Kronecker delta function. • If the constants Kn are all equal to 1 then the ϕn(t) are said to be orthonormal functions.

3. Example 2.11 Orthogonal Complex Exponential Functions

4. Orthogonal Series • Theorem:Assume w(t) represents a waveform over the interval a < t <b. Then w(t) can be represented over the interval (a, b)by the series where, the coefficients an are given by following where n is an integer value : • If w(t) can be represented without any errors in this way we call the set of functions {φn} as a “Complete Set” • Examples for complete sets: • Harmonic Sinusoidal Sets {Sin(nw0t)} • Complex Expoents {ejnwt} • Bessel Functions • Legendare polynominals

5. Orthogonal Series Proof of theorem: Assume that the set {φn} is sufficient to represent the waveform w(t) over the interval a < t <b by the series We operate the integral operator on both sides to get, • Now, since we can find the coefficients an writing w(t) in series form is possible. Thus theorem is proved.

6. Application of Orthogonal Series • It is also possible to generate w(t) from the ϕj(t) functions and the coefficients aj. • In this case, w(t) is approximated by using a reasonable number of the ϕj(t) functions. w(t) is realized by adding weighted versions of orthogonal functions

7. Ex. Square Waves Using Sine Waves. n =1 n =3 n =5 http://www.educatorscorner.com/index.cgi?CONTENT_ID=2487

8. Fourier SeriesComplex Fourier Series • The frequency f0 = 1/T0 is said to be the fundamental frequencyand the frequency nf0is said to be the nth harmonic frequency, when n>1.

9. Some Properties of Complex Fourier Series

10. Some Properties of Complex Fourier Series

11. Quadrature Fourier Series • The Quadrature Form of the Fourier series representing any physical waveform w(t) over the interval a < t < a+T0 is, where the orthogonal functions are cos(nω0t) and sin(nω0t). Using we can find the Fourier coefficients as:

12. Since these sinusoidal orthogonal functions are periodic, this series is periodic with the fundamental period T0. • The Complex Fourier Series, and the Quadrature Fourier Series are equivalent representations. • This can be shown by expressing the complex number cn as below Quadrature Fourier Series For all integer values of n and Thus we obtain the identities and

13. Polar Fourier Series • The POLAR F Form is where w(t)is real and The above two equations may be inverted, and we obtain

14. Polar Fourier Series Coefficients

15. Theorem:If a waveform is periodic with period T0, the spectrum of the waveform w(t) is Line Spetra for Periodic Waveforms where f0 = 1/T0 and cn are the phasor Fourier coefficients of the waveform Proof: Taking the Fourier transform of both sides, we obtain Here the integral representation for a delta function was used.

16. Line Spectra for Periodic Waveforms Theorem:If w(t) is a periodic function with period T0 and is represented by Where, then the Fourier coefficients are given by: The Fourier Series Coefficients can also be calculated from the periodic sample values of the Fourier Transform.

17. Line Spectra for Periodic Waveforms Line Spectra for Periodic Waveforms h(t) The Fourier Series Coefficients of the periodic signal can be calculated from the Fourier Transform of the similar nonperiodic signal. The sample values for the Fourier transform gives the Fourier series coefficients.

18. Line Spectra for Periodic Waveforms Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum

19. Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

20. Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave  T Sa(fT) • Now evaluate the coefficients from the Fourier Transform • Now compare the spectrum for this periodic rectangular wave (solid lines) with the spectrum for the rectangular pulse. • Note that the spectrum for the periodic wave contains spectral lines, whereas the spectrum for the nonperiodic pulse is continuous. • Note that the envelope of the spectrum for both cases is the same |(sin x)/x| shape, where x=Tf. • Consequently, the Null Bandwidth (for the envelope) is 1/T for both cases, where T is the pulse width. • This is a basic property of digital signaling with rectangular pulse shapes. The null bandwidth is the reciprocal of the pulse width.

21. Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum

22. Normalized Power Theorem:For a periodic waveform w(t), the normalized power is givenby: where the {cn} are the complex Fourier coefficients for the waveform. Proof: For periodic w(t), the Fourier series representation is valid over all time and may be substituted into Eq.(2-12) to evaluate the normalized power:

23. Power Spectral Density for Periodic Waveforms Theorem:For a periodic waveform, the power spectral density (PSD) is given by where T0 = 1/f0 is the period of the waveform and {cn} are the corresponding Fourier coefficients for the waveform. PSD is the FT of the Autocorrelation function

24. The PSD for the periodic square wave will be found. • Because the waveform is periodic, FS coefficients can be used to evaluate the PSD. Consequently this problem becomes one of evaluating the FS coefficients. Power Spectral Density for a Square Wave