1. MEASUREMENTS IN FREQUENCY DOMAIN: GENERAL ASPECTS

2. TOPICS Going beyond time Foundations of Fourier Theory Importance and usage of Decibel

3. GOING BEYOND TIME (frequency domain)

4. Any real signal can be obtained through the sum of sinusoidal components. Going beyond time…...

5. Mathematical model of a sinusoidal component c(t) : c(t) = A cos(2πf t + φ) A : amplitude f : frequency φ : phase (constant)

6. Tridimensional representation: time, frequency, amplitude

7. Why frequency domain measurements ? Time: problem Frequency: solution

8. Some fields of application Telecommunications, Telematics Analog and Power Electronics Electromagnetics Electrical Systems and Machines Bioengineering

9. Measurement examples Modulation Distorsion Noise

10. FOUNDATIONS OF FOURIER THEORY Jean Baptiste Joseph Fourier, 1768-1830, French mathematician and physicist.

11. Fourier Series Periodic and time-continuous signal Expansion in series of sinusoidal functions

12. Fourier Series c n = spectrum of s(t) amplitude = │c n │; phase = arg(c n ) period

13. Fourier Transform Aperiodic and time-continuous signal Analysis S(f) = spectrum of s(t) Synthesis

14. Spectrum examples (periodic signals) Time T Sinusoidal wave Frequency f = 1/T T Time Square wave Frequency 1/T3/T5/T

15. Spectrum examples (aperiodic signals) Frequency Time Transient signal Frequency Time Ideal impulse

16. IMPORTANCE AND USAGE OF DECIBEL

17. Decibel definition Ratio between two power values: P 2 is expressed in relative terms with respect to P 1 Swapping P 2 and P 1 changes the sign of A (dB)

18. Role of resistances If the powers refer to a couple of voltages applied to two resistances: V 2 is the rms voltage applied to R 2 V 1 is the rms voltage applied to R 1

19. Role of resistances If R 2 = R 1

20. Fundamentals rules  Inversion  Addition  Subtraction

21. Absolute values in Decibel : dBm Choosing 1mW as reference power value P (mW) is expressed in milliwatt P (dBm) is independent of R

22. V REF is the rms voltage producing 1mW power: Absolute values in Decibel : dBm

23. For R = 50 Ω, V REF = 0.2236 For R = 75 Ω, V REF = 0.2739

24. Absolute values in Decibel : dBV Choosing 1V rms as reference voltage value V (dBV) is indipendent of R

25. dBm/dBV conversion R = 50 Ω R = 75 Ω