1. 1 5.3. Noise characteristics Reference: [4] The signal-to-noise ratio is the measure for the extent to which a signal can be distinguished from the background noise: SNR msr . S msr N msr References: [1] and [2] It is assumed that the signal power, S msr, and the noise power, N msr, are dissipated in the noiseless input impedance of the measurement system. 5.3.1. Signal-to-noise ratio, SNR 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR

2. 2 Example: Calculation of the signal-to-noise ratio at the (noiseless) input of a measurement system Z s = R s + X s Z msr = R msr + X msr Measurement objectMeasurement system V in 1) S in , V in rms 2 R in  Z s + Z in  2 2) N in , V n rms 2 R in  Z s + Z in  2 Noiseless V msr 3) SNR in  V in rms 2 V n rms 2 V in rms 2 4 k T R   f n 

3. 3 The concept of noise factor and noise figure was developed in the 1940s. In spite of several serious limitations, this concept is still widely used today. (About the limitation of the noise factor concept, we will talk in the end of this Section.) Meanwhile, let us start from definitions. 5.3.2. Noise factor, F, and noise figure, NF 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Reference: [2]

4. 4 The noise factor, F, compares the noise performance of a device (measurement system) to that of an ideal (noiseless) device: F , NoNo*NoNo* 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF where N o is the noise power at the output of the noisy measurement system, with the noisy measurement signal connected to the input, and N o * is the noise power at the output of the same system, which is now considered to be free of noise. The output noise then comes only from the measured input signal. References: [1] and [2] A. Definitions

5. 5 The noise power of an ideal (noiseless) measurement system is due to the thermal noise power of the source resistance. Therefore, the noise factor can also be written as F , N o N os 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF where N os is the contribution to the output noise power due to the source noise. References: [2]

6. 6 An equivalent definition of noise factor is the input SNR divided by the output SNR References: [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF where A P is the power amplification, defined as S o /S msr. F F  N o N o * F , SNR msr SNR o N o S o N o * S o  N o S msr A P N msr A P S o  S msr /N msr S o /N o 

7. 7 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF RsRs Measurement objectMeasurement system RLRL G e ns F  NoNo*NoNo* V no 2 /R L 4 kTR s B n (G A V ) 2 /R L  V no 2 4 kTR s B n (G A V ) 2  VoVo Voltage gain, A V V msr B. Calculation of noise factor. Example

8. 8 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF F  V no 2 4 kTR s B n (G A V ) 2 The following three characteristics of noise factor can be seen by examining the obtained equation: 1. It is independent of load resistance R L, 2. It does depend on source resistance R s, 3.If the measurement system were completely noiseless, the noise factor would equal one. References: [2] Conclusions:

9. 9 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Noise factor expressed in decibels is called noise figure (NF) : References: [2] NF  10 log F. Due to the bandwidth term in the denominator there are two ways to specify the noise factor: (1) a spot noise, measured at specified frequency over a 1  Hz bandwidth,or (2) an integrated, or average noise measured over a specified bandwidth. C. Noise figure F  V no 2 4 kTR s B n (G A V ) 2

10. 10 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] We will consider the following methods for the measurement of noise factor: (1) the single-frequency method, and (2) the white noise method. E. Measurement of noise factor 1) Single-frequency method. According to this method, a sinusoidal test signal V in (rms) is increased until the output power doubles. Under this condition the following equation is satisfied: RsRs Measurement objectMeasurement system RLRL V in VoVo Voltage gain, A V V msr G

11. 11 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] RsRs Measurement objectMeasurement system RLRL V in VoVo Voltage gain, A V V msr 1) (V in G A V ) 2 + V no 2  2 V no 2 V in  0 2) V no 2  (V in G A V ) 2 V in  0 3) F  No*No* V no 2 V in  0 (V in G A V ) 2 4 kTR s B n (G A V ) 2  V in 2 4 kTR s B n  G

12. 12 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] F V in 2 4 kTR s B n  The disadvantage of the single-frequency meted is that the noise bandwidth of the measurement system must be known. A better method of measuring noise factor is to use a white noise source. 2) White noise method. This method is similar to the previous one. The only difference is that the sinusoidal signal generator is now replaced with a white noise current source:

13. 13 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Measurement objectMeasurement system RLRL i in ( f ) VoVo Voltage gain, A V V in 1) (i in R s G A V ) 2 B n + V no 2  2 V no 2 i t  0 2) V no 2  (i in R s G A V ) 2 B n i t  0 3) F  No*No* V no 2 i t  0 (i in R s G A V ) 2 B n 4 kTR s B n (G A V ) 2  i in 2 R s 4 kT  RsRs G

14. 14 i in 2 R s 4 kT  5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF F The noise factor is now a function of only the test noise signal, the value of the source resistance, and temperature. All of these quantities are easily measured. Neither the gain nor the noise bandwidth of the measurement system need be known.

15. 15 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Calculating SNR and input noise voltage from NF. 5.3.3. Calculating SNR and input noise voltage from NF Reference: [2]

16. 16 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF 1. Increasing the source resistance may decrease the noise factor, while increasing the total noise in the circuit. 2. If a purely reactive source is used, noise factor is meaningless, since the source noise is zero, making the noise actor infinite. 3.When the measurement system noise is only a small part of of the source thermal noise (as with some low-noise FETs), the noise factor requires taking the ratio of two almost equal numbers. this can produce inaccurate results. References: [2] The concept of noise factor has three major limitations: D. Limitations of the noise factor concept F  V no 2 4 kTR s B n (G A V ) 2 The concept of noise factor has three major limitations:

17. 17 A direct comparison of two noise factors is only meaningful if both are measured at the same source resistance. Noise factors varies with the bias conditions, frequency, and temperature as well as source resistance, and all of these should be defined when specifying noise factor. Knowing the noise factor for one value of source does not allow the calculation of the noise factor at other values of resistance. This is because both the source noise and measurement system noise vary as the source resistance is changed. 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] Noise factor is usually specified for matched devices and is a popular figure of merit in RF applications.

18. 18 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model Reference: [2] 5.3.4. Two noise source model A more recent (1956) approach and one that overcomes the limitations of noise factor, is to model the noise in terms of an equivalent noise voltage and current. The actual network can be modeled as a noise-free network with two noise generators, e n and i n, connected to its input: RsRs Measurement objectMeasurement system RLRL G V in VoVo V msr inin R msr Noiseless AVAV enen

19. 19 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model Reference: [2] The e n source represents the network noise that exists when R s equals zero, and the i n source represents the additional noise that occurs when R s does not equal zero, The use of these two noise generators plus a complex correlation coefficient completely characterizes the noise performance of the network. At relatively low frequencies, the correlation between the e n and i n noise sources can be neglected. RsRs Measurement objectMeasurement system RLRL G V in VoVo V msr enen inin R msr Noiseless AVAV

20. 20 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model A. Measurement of e n and i n

21. 21 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model Reference: www.analog.com Example: Input voltage and current noise spectra (ultralow noise, high speed, BiFET op-amp AD745) enen inin

22. 22 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model Assuming no correlation between the noise sources, the total equivalent input noise voltage of the whole system can be found by superposition. RsRs Measurement objectMeasurement system RLRL G V in VoVo V msr enen inin R msr Noiseless AVAV B. Total input noise as a function of the source impedance

23. 23 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. RsRs Measurement objectMeasurement system RLRL G V in VoVo V msr enen inin R msr Noiseless AVAV GiGi + [I n rms R s R msr /(R s +R msr ) A V ] 2, 2) V n in rms 2 =V o 2 / (G A V ) 2, + [V n rms G A V ] 2 1) V o rms 2 = 4 kTR s B [ R msr /(R s +R msr ) A V ] 2 G

24. 24 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. Example: Typical total input equivalent noise voltage as a function of R s 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 v n in rms, nV/Hz 0.5 e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 enen  4kTR s B R s,  i n R s

25. 25 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.4. Two noise source model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. RsRs Measurement objectMeasurement system RLRL G V in VoVo Voltage gain, A V V msr We now can connect an equivalent noise generator in series with input signal voltage source to model the total input voltage of the whole system. V n in

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