ECE 590 Microwave Transmission for Telecommunications

1. ECE 590Microwave Transmission for Telecommunications Noise and Distortion in Microwave Systems March 18, 25, 2004

2. Random Processes

3. Random Processes

4. Expected Values

5. Expected Values

6. Expected Values

7. Autocorrelation and Power Spectral Density

8. Autocorrelation and Power Spectral Density

9. Noise in Microwave Circuits • Result of random motions of charges or charge carriers in devices and materials • Thermal noise (most basic type) • thermal vibration of bound charges (also called Johnson or Nyquist noise) • Shot noise • random fluctuations of charge carriers • Flicker noise • occurs in solid-state components and varies inversely with frequency (1/f -noise)

10. Noise in Microwave Circuits • Plasma noise • random motion of charges in ionized gas such as a plasma, the ionosphere, or sparking electrical contacts • Quantum noise • results from the quantized nature of charge carriers and photons; often insignificant relative to other noise sources

11. Noise power and Equivalent Noise Temperature

12. Noise power and Equivalent Noise Temperature

13. Noise in Linear Systems

14. Noise in Linear Systems

15. Gaussian white noise through an ideal low-pass filter

16. Gaussian white noise through an ideal integrator

17. Mixing of noise: frequency conversion

18. Mixing of noise: frequency conversion

19. Basic Threshold Detection

20. Basic Threshold Detection

21. Graphical Representation of Probability of Error for Basic Threshold Detection

23. Noise Temperature and Noise Figure

24. Noise Figure Noisy Rf and microwave components can be characterized by an equivalent noise temperature. An alternative is the noise figure which is the degradation of the signal to noise ratio between the input and the output of the component, or F = (Si/Ni)/ (S0/N0)  1. The input noise power, Ni = k T0 B; Pi= Si+ Ni ; P0= S0+ N0; S0= G Si; N0= kGB(T0+ Te) ;

25. Noise Figure So F = [(Si/ k T0 B)]/ [(G Si / k G B (T0+ Te)] =(T0+ Te)/ T0 = 1 + Te/ T0  1. Or the temperature of the noisy network Te = (F - 1) T0 . Let Nadded = noise power added by the network, the output noise power, N0= G (Ni+ Nadded) So F = [(Si/ Ni)]/ [(G Si / G (Ni+ Nadded)] = 1 + Nadded/ Ni

26. Noise Figure of a Lossy Line Lossy transmission line (attenuator) held at a physical temperature, T. Power Gain, G<1 so power loss factor = L =1/G>1 If the line input is terminated with a matched load at temperature T, then the output will appear as a resistor of value R and temperature T. Output Noise power is the sum of the input noise power attenuated through the lossy line plus the noise power added by the lossy line itself .

27. Noise Figure of a Lossy Line So the output Noise power, No = kTB = G(kTB + Nadded), where Nadded is the noise generated by the line. Therefore, Nadded = {(1/G) - 1 }kTB = (L-1) kTB The equivalent noise temperature Te of the lossy line becomes: Te = Nadded / KB = (L - 1) T; and the noise figure is F = 1 + Te / T0 = 1 + (L - 1) T / T0

28. Noise Figure of Cascaded Components Consider a cascade of two components having power gains G1 and G2, noise figures F1 and F2 and noise temperatures T1 and T2. Find overall noise figure, T and noise temperature T of the cascade as if it were the single component with Ni = k T0 B. Using noise temperatures, the noise power at the output of the first stage is N1= G1 k B T0+ G1 k B Te1; and the output at the second is N0= G2 N1+ G2 k B Te2 = G1 G2 k B (T0 + Te1 + Te2 / G1)

29. Noise Figure of Cascaded Components For the equivalent single system: N0= G1 G2 k B (T0 + Te) So the noise of the cascade system is Te = Te1 + Te2 / G1 Recall F = 1 + Te/ T0 so the cascade system F = 1+ Te1/ T0 + Te2 / (G1 T0) = F1 + ( F2 - 1) / G1; more generally Te = Te1 + Te2 / G1 + Te3 / (G1G1) F = F1 + ( F2 - 1) / G1+ ( F3 - 1) / G1 G2

33. Noise Figure of a Passive Two-Port Network Impedance mismatches may be defined at each port in terms of the reflection coefficients,  as shown in the diagram. Assume the network is at temperature, T and the input noise power is N1 = k T B is applied to the input of the network. The available output noise at port 2 is N2 = G21 k T B + G21 Nadded the noise generated internally by the network (referenced at port 1). G21 is the available gain of the network from port 1 to port 2.

34. Noise Figure of a Passive Two-Port Network The available gain can be expressed in terms of the S-parameters of the network and the port mismatches as G21 = power available from network divided by power available from source = { |S21|2 (1- | s | 2)}/ | 1+S11s | 2(1- | out | 2) and the output mismatch is out = S22+ S12S21s /(1- S11s ) From N2=k T B, find Nadded = (1/G21-1)k T B, and the equivalent noise temperature is Te = Nadded /kB = T(1- G21)/ G21, and F = (1/G21-1)T/T0 Can apply to examples mismatched lossy line and Wilkinson power divider.

35. Gain Compression General non-linear network with an input voltage vi and and output voltage v0 can be expressed in a Taylor series expansion: v0 = a0 + a1vi + a2vi2 + a3vi3 + … where the Taylor coefficients are given by: a0 = v0 (0) (DC output); {rectifier converting ac to dc} a1 = dv0 / dvi| vi =0 (linear output) ; {linear attenuator or amplifier} a2 = d2v0 / dvi2| vi =0 (squared output) ; {mixing and other frequency conversion functions}

36. Gain Compression Let vi = V0 cos 0t then evaluate v0 = a0 + a1vi + a2vi2 + a3vi3 + … v0 = a0 + a1 V0 cos 0t + a2 V0 2 cos 20t + a3 V0 3 cos 30t + … =( a0 + ½ a2 V0 2 ) + (a1 V0 + ¾ a3 V0 3 ) cos 0t + ½ a2 V0 2 cos 20t + ¼ a3 V0 3 cos 30t + … This result leads to the voltage gain of the signal component at frequency 0 Gv = v0 (0 )/ vi (0 ) = (a1 V0 + ¾ a3 V0 3 ) / V0 = a1 + ¾ a3 V0 2 (retaining only terms through the third order)

37. Gain Compression Gv = v0 (0)/ vi (0)= (a1 V0 + ¾ a3 V0 3 ) / V0 = a1 + ¾ a3 V0 2 here we see the a1 term plus a term proportional to the square of the magnitude of the amplitude of the input voltage. The coefficient a3 is typically negative; so the gain of the amplifier tens to decrease for large values of V0. This is gain compression or saturation.

38. Intermodulation Distortion For a single input frequency, or tone, 0, the output will consist of harmonics of the input signal of the form, n 0, for n = 0, 1, 2, …. Usually these harmonics are out of the passband of the amplifier, but that is not true when the input consists of two closely spaced frequencies. Let vi = V0(cos 1t + cos 2t ); where 1 ~ 2. Recall v0 = a0 + a1vi + a2vi2 + a3vi3 + … ; hence

39. Intermodulation Distortion The output spectrum consists of harmonics of the form, m1+n2 with m, n = 0, 1, 2, 3, … These combinations of the two input frequencies are call intermodulation products, with order |m| + |n|. Generally, they are undesirable; however, in cases, for example a mixer, the the sum or difference frequencies form the desired outputs. Note that they are both far from 1 and 2. But the terms 21 - 2 and 22 - 1 are close to 1 and 2. Which causes third-order intermodulation distortion.

40. Third-Order Intercept Point Plot of first and third-order products of the output versus input power on a log-log plot hence the slopes represent the powers.

41. Dynamic Range