1. Eeng 360 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. Eeng 360 2 Orthogonal Functions Orthogonal Functions  Definition : Functions ϕ n (t) and ϕ m (t) are said to be Orthogonal with respect to each other the interval a < t < b if they satisfy the condition, where δ nm is called the Kronecker delta function. If the constants K n are all equal to 1 then the ϕ n (t) are said to be orthonormal functions.

3. Eeng 360 3 Example 2.11 Orthogonal Complex Exponential Functions

4. Eeng 360 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 a n 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 {φ n } as a “Complete Set” Examples for complete sets: Harmonic Sinusoidal Sets {Sin(nw 0 t)} Complex Expoents {e jnwt } Bessel Functions Legendare polynominals

5. Eeng 360 5 Orthogonal Series Proof of theorem: Assume that the set {φ n } 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 t o get, Now, since we can find the coefficients a n writing w(t) in series form is possible. Thus theorem is proved.

6. Eeng 360 6 Application of Orthogonal Series  It is also possible to generate w(t) from the ϕ j (t) functions and the coefficients a j.  In this case, w(t) is approximated by using a reasonable number of the ϕ j (t) functions. w(t) is realized by adding weighted versions of orthogonal functions

7. Eeng 360 7 Ex. Square Waves Using Sine Waves. http://www.educatorscorner.com/index.cgi?CONTENT_ID=2487 n =1 n =3 n =5

8. Eeng 360 8 Fourier Series Complex Fourier Series  The frequency f 0 = 1/T 0 is said to be the fundamental frequency and the frequency nf 0 is said to be the nth harmonic frequency, when n>1.

9. Eeng 360 9 Some Properties of Complex Fourier Series

10. Eeng 360 10 Some Properties of Complex Fourier Series

11. Eeng 360 11 Quadrature Fourier Series  The Quadrature Form of the Fourier series representing any physical waveform w(t) over the interval a < t < a+T 0 is, where the orthogonal functions are cos(nω 0 t) and sin(nω 0 t). Using we can find the Fourier coefficients as:

12. Eeng 360 12 Quadrature Fourier Series Since these sinusoidal orthogonal functions are periodic, this series is periodic with the fundamental period T 0. The Complex Fourier Series, and the Quadrature Fourier Series are equivalent representations. This can be shown by expressing the complex number c n as below For all integer values of n and Thus we obtain the identities and

13. Eeng 360 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. Eeng 360 14 Polar Fourier Series Coefficients

15. Eeng 360 15 Line Spetra for Periodic Waveforms Theorem: If a waveform is periodic with period T 0, the spectrum of the waveform w(t) is where f 0 = 1/T 0 and c n 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. Eeng 360 16 Line Spectra for Periodic Waveforms Theorem: If w(t) is a periodic function with period T 0 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. Eeng 360 17 Line Spectra for Periodic Waveforms h(t)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. Eeng 360 18 Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum Line Spectra for Periodic Waveforms

19. Eeng 360 19 Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

20. Eeng 360 20  T Sa(  fT) 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.  Now evaluate the coefficients from the Fourier Transform Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

21. Eeng 360 21 Single Pulse Continous Spectrum Periodic Pulse Train Line Spectrum Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave

22. Eeng 360 22 Normalized Power Theorem: For a periodic waveform w(t), the normalized power is given by: where the {c n } 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. Eeng 360 23 Power Spectral Density for Periodic Waveforms Theorem: For a periodic waveform, the power spectral density (PSD) is given by where T 0 = 1/f 0 is the period of the waveform and {c n } are the corresponding Fourier coefficients for the waveform. PSD is the FT of the Autocorrelation function

24. Eeng 360 24 Power Spectral Density for a Square Wave 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.