Equivalent Noise Bandwidth
4.20 Narrowband Band-pass Noise
Band-pass filtering of signals arises in many situations, the basic arrangement being shown in Fig. 4.20.1. The fil- ter has an equivalent noise bandwidth BN(see Section 4.2) and a center frequency fc. A narrowband system is one in which the center frequency is much greater than the bandwidth, which is the situation to be considered here.
The signal source is shown as a voltage generator of internal resistance Rs. System noise is referred to the input as a thermal noise source at a noise temperature Ts. The available power spectral density is, from Eq. (4.2.9),
(4.20.1) For the ideal band-pass system shown, the spectral density is not altered by transmission through the fil- ter, but the filter bandwidth determines the available noise power as kTsBN. So far, this is a result that has already been encountered in general. An alternative description of the output noise, however, turns out to be very useful, especially in connection with the modulation systems described in later chapters. The waveforms of input and output noise voltages are shown in Fig. 4.20.2.
The output waveform has the form of a modulated wave and can be expressed mathematically as (4.20.2) n(t)An(t) cos(ct n(t))
Ga(f)kTs F ENR
Y1
SOLUTION The excess noise ratio, as a power ratio, is ENR 101.425.12
The Yfactor expressed as a power ratio is Y101.97.94. From the definition of ENR, the hot temperature is
Substituting this in Eq. (4.19.5) gives
760 K
Te 75757.94290 7.941
ThTo(ENR1)290(25.121) 7575 K
Figure 4.20.1 Noise in a band-pass system.
Rs BPF
Ga =kTs n(t)
This represents the noise in terms of a randomly varying voltage envelope An(t) and a random phase angle n(t). These components are readily identified as part of the waveform, as shown in Fig. 4.20.2, but an equiv- alent although not so apparent expression can be obtained by trigonometric expansion of the output wave- form as
(4.20.3) Here,nI(t) is a random noise voltage termed the in-phase componentbecause it multiplies a cosine term used as a reference phasor, and nQ(t) is a similar random voltage termed the quadrature componentbecause it multiplies a sine term, which is therefore 90out of phase, or in quadrature with, the reference phasor. The reason for using this form of equation is that, when dealing with modulated signals, the output noise voltage is determined by these two components (this is described in detail in later chapters on modulation). The two noise voltages nI(t) and nQ(t) appear to modulate a carrier at frequency fcand are known as the low-pass equivalent noise voltages.The carrier fcmay be chosen anywhere within the passband, but the analysis is simplified by placing it at the center as shown. This is illustrated in Fig. 4.20.3.
A number of important relationships exist between nI(t) and nQ(t) and n(t), some of which will be stated here without proof. All three have similar noise characteristics and nI(t) and nQ(t) are uncorrected.
Of particular importance in later work on modulation is that where the power spectral density of n(t) is Ga(f)kTsthe power spectral densities for nI(t) and nQ(t) are
(4.20.4) This important result, which is illustrated in Fig. 4.20.3, will be encountered again in relation to modulated signals.
GI(f)GQ(f)2kTs n(t)nI(t) cos ctnQ(t) sin ct
Figure 4.20.3 Noise spectral densities.
2kTs GI(f) = GQ(f)
Ga(f) = kTs
0 W fc– W fc fc+ W
Figure 4.20.2 Input and output noise waveforms for a band-pass system.
BPF Noise input
Randomly phased crossovers, φn(t) n(t)
t t
An(t)
PROBLEMS
Assume that qe1.6 1019C,k1.38 1023J/K and room temperature To290 K applies unless otherwise stated.
4.1. Explain how thermal noise power varies (a) with temperature and (b) with frequency bandwidth.
Thermal noise from a resistor is measured as 4 1017W for a given bandwidth and at a tempera- ture of 20C. What will the noise power be when the temperature is changed to (c) 50C; (d) 70 K?
4.2. Given two resistors R110 kand R215 k, calculate the thermal noise voltage generated by (a) R1, (b) R2, (c) R1 in series with R2, and (d) R1 in parallel with R2. Assume a 20-MHz noise bandwidth.
4.3. Three resistors have values R110 k,R214 k, and R324 k. It is known that the thermal noise voltage generated by R1is 0.3 V. Calculate the thermal noise voltage generated by (a) the three resistors connected in series and (b) connected in parallel.
4.4. A 50- source is connected to a T-attenuator, the two series resistors [R1and R2of Fig. 1.2.2(b)] each being 100 , and the central parallel resistor [R3of Fig. 1.2.2.(b)] being 150 . Calculate the noise voltage appearing at the output terminals for a noise bandwidth of 1 MHZ.
4.5. The noise generated by a 1000-resistor can be represented by a 4-nA current source in parallel with the 1000 . Determine the equivalent emf source representation.
4.6. Explain why inductance and capacitance do not generate noise.
4.7. The available noise power spectral density at the input to an LCfilter is kTojoules. The filter has a transfer function that can be approximated by
Calculate the available output noise power.
4.8. A 100-kresistor at room temperature is placed at the terminals of the filter in Problem 4.7. Assuming that the resistor does not alter the response curve, calculate the mean-square voltage at the output.
4.9. Explain how a capacitance Cconnected across a resistor Raffects the thermal noise appearing at the terminals. What is the effective noise bandwidth of a 100-pF capacitor connected in parallel with a 10-kresistor?
4.10. A 100-kresistor is connected in parallel with a 100-pF capacitor. Determine the effective noise bandwidth and the noise voltage appearing at the terminals of the combination.
4.11. Calculate the equivalent noise bandwidth for the filter of Problem 4.7.
4.12. A resistor has a self-capacitance of 3 pF. Calculate the mean-square noise voltage at its terminals at room temperature.
4.13. Determine the mean-square noise voltage at the terminals of the resistor in Problem 4.12 if it is assumed that the self-capacitance is zero. Discuss the validity of this result.
4.14. A single tuned circuit has a Q-factor of 70 and is resonant at 3 MHz with a 470 pF tuning capacitor.
Calculate the equivalent noise bandwidth for the circuit.
f 35 kHz 0
15 f35 kHz 1.75 f
2104
0f15 kHz H(f) 1
4.15. The tuned circuit of Problem 4.14 is fed from a current source at resonance, the internal resistance of the source being equal to the dynamic impedance of the circuit. Calculate the equivalent noise band- width in this case.
4.16. A signal source having an internal resistance of 50 is connected to a tap on the inductor of a tuned circuit, the tapping point being one-quarter up from the ground connection. The undamped Q-factor of the circuit is 75, and the tuning capacitance is 900 pF for resonance at 500 kHz. Calculate the equivalent noise bandwidth of the system, assuming that the tapping ratio can be treated as an ideal transformer coupling.
4.17. Write brief notes on the sources of noise, other than thermal, that arise in electronic equipment.
Describe how the power spectral density varies with frequency in each case.
4.18. Calculate the shot noise current present on a direct current of 13 mA for a noise bandwidth of 7 MHz.
4.19. Calculate the mean-square spectral density of shot noise current accompanying a direct current of 5 mA.
4.20. An amplifier has an equivalent noise resistance of 300 and an equivalent shot-noise current of 200A. It is fed from a signal source that has an internal resistance of 50 . Calculate the total noise voltage at the amplifier input for a noise bandwidth of 1 MHz.
4.21. Repeat Problem 4.20 for a source resistance of 600 .
4.22. A signal source has an emf of 3 V and an internal resistance of 450 . It is connected to an ampli- fier that has an equivalent noise resistance of 250 and an equivalent shot noise current of 300 A.
Calculate the S/N ratio in decibels at the input. The equivalent noise bandwidth is 10 kHz.
4.23. An emf source of 1 V rms has an internal resistance of 600 . Calculate the S/N ratio at its terminals. Calculate the new S/N ratio at the terminals when the source is connected to a 600-load.
4.24. A telephone transmission system has three identical links, each having an S/N ratio of 50 dB. Calculate the output S/N ratio of the system. One of the links develops a fault that reduces its S/N ratio to 47 dB.
Calculate the output S/N under these circumstances.
4.25. Three telephone circuits, each having an S/N ratio of 44 dB, are connected in tandem. Determine the overall S/N ratio. A fourth circuit is now added that has an S/N ratio of 34 dB. Determine the new overall value.
4.26. Define noise factorin terms of input and output signal-to-noise ratios of a network. The noise factor of an amplifier is given as 51. If the input S/N is 50 dB, calculate the output S/N ratio in decibels.
4.27. The noise factor of a radio receiver is 151. Calculate its noise figure. Determine the output S/N ratio when the input S/N ratio to the receiver is 35 dB.
4.28. The noise figure of an amplifier is 11 dB. Determine the fraction of the total available noise power contributed by the amplifier, referred to the input.
4.29. The available output noise power from an amplifier is 100 nW, the available power gain of the amplifier is 50 dB, and the equivalent noise bandwidth is 30 MHz. Calculate the noise figure.
4.30. The noise figure of an amplifier is 7 dB. Calculate the equivalent amplifier noise referred to the input for a bandwidth of 500 MHz.
4.31. Three amplifiers 1, 2, and 3 have the following characteristics:
F19 dB, G148 dB F26 dB, G235 dB F34 dB, G320 dB
The amplifiers are connected in tandem. Determine which combination gives the lowest noise factor referred to the input.
4.32. An amplifier has an equivalent noise resistance of 350 Ωand an equivalent shot noise current of 400 A.
It is fed from a 1000-source. Calculate the noise factor.
4.33. Calculate the optimum source resistance for the amplifier in Problem 4.32.
4.34. A source has an internal resistance of 50 and is to be coupled into an amplifier input for which the equivalent noise generators are Rn500 and IEQ300 A. Calculate the turns ratio of the input transformer required to minimize the noise factor, assuming that impedance is transformed according to the square of the turns ratio.
4.35. A signal source is connected to an amplifier through a cable that is matched, but that introduces a loss of 2.3 dB. What is the noise factor of the cable?
4.36. An attenuator has an insertion loss IL 0.24. Determine its noise figure.
4.37. An amplifier has a noise figure of 7 dB. Calculate its equivalent noise temperature.
4.38. A cable has an insertion loss of 2 dB. Determine its equivalent noise temperature referred to the input.
4.39. A cable that has a power loss of 3 dB is connected to the input of an amplifier, which has a noise tem- perature of 200 K. Calculate the overall noise temperature referred to the cable input.
4.40. A mixer circuit has a noise figure of 12 dB. It is preceded by an amplifier that has an equivalent noise temperature of 200 K and a power gain of 30 dB. Calculate the equivalent noise temperature of the combination referred to the amplifier input.
4.41. An amplifier has a gain of 12 dB and a noise temperature of 120 K. The amplifier may be connected into a receiving system at either end of the cable feeding the main receiver from the antenna. The cable has an insertion loss of 12 dB. Determine which connection gives the lowest overall noise figure. The noise characteristics of the main receiver may be ignored.
4.42. A satellite receiving system consists of a low noise amplifier (LNA) that has a gain of 47 dB and a noise temperature of 120 K, a cable with a loss of 6.5 dB, and a main receiver with a noise factor of 9 dB. Calculate the equivalent noise temperature of the overall system referred to the input for the following system connections: (a) the LNA at the input, followed by the cable connecting to the main receiver; (b) the input direct to the cable, which then connects to the LNA, which in turn is connected directly to the main receiver.
4.43. Using the information given in the text, derive Eq. (4.18.4). Two amplifiers are connected in cascade.
The first amplifier has a noise temperature of 120 K and a power gain of 15 dB; the second, a noise temperature of 300 K. Calculate the overall noise temperature of the cascaded connection referred to the input.
4.44. The ENR for an avalanche diode is 13 dB. Given that the cold temperature is equal to room temper- ature (290 K), determine the hot temperature of the source.
4.45. In a noise measurement using an avalanche diode, the ENR was 14.3 dB. The noise power output with the diode on was 45 dBm, and with the diode off, 36 dBm. Calculate the noise temperature of the device under test.
4.46. Derive Eq. (4.19.6) of the text. In the measurement of noise factor, the ENR 13.7 dB and Y7 dB.
Calculate the noise figure and the equivalent noise temperature of the device under test.
4.47. Explore MATLAB functions for generating random sequences. (Hint:rand(.), and randn(.) functions.) 4.48. Plot the thermal noise voltage generated by a 10kresistor, when the temperature is varied from 0K
to 300K. Let the bandwidth of interest be 10MHz. Use MATLAB/Mathematica.
4.49. Explore the MATLAB functions to compute the power spectral density (PSD) of a noise signal.
4.50. Write a MATLAB program to simulate the rolling of a fair die.(Hint: Use randperm(6).)
4.51. Generate a random noise vector containing 20 elements, with zero mean and variance 4. (Hint: Use randn(1,20)*2)
4.52. Show that mean(randn(1,x)) →0 as x → ∞.
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