Noise figure calculator: cascade, noise temperature and sensitivity

What noise figure is: noise factor, noise figure and noise temperature

Every stage of a receiver adds noise of its own to the noise that arrives with the signal. Noise figure is the measure of how much. Keysight's application note AN 57-1, Fundamentals of RF and Microwave Noise Figure Measurements, gives Friis's definition: "the noise figure F of a network to be the ratio of the signal-to-noise power ratio at the input to the signal-to-noise power ratio at the output" (p. 7). A perfect amplifier would raise signal and noise together and leave the ratio alone; a real one adds noise from its own components and the ratio falls. The same page is explicit that this is not a statement about gain: "Once noise is added to the signal, subsequent gain amplifies signal and noise together and does not change the signal-to-noise ratio."

The ratio depends on how much noise was at the input to begin with, so the definition fixes it. Friis "suggested a reference source temperature of 290K", and at that temperature the available noise density kT₀ is, in AN 57-1's words, "the even number 4.00 x 10-21 watts per hertz of bandwidth (–174 dBm/Hz)" (p. 8). With the exact SI value of Boltzmann's constant, k = 1.380 649 × 10⁻²³ J/K, it is 4.004 × 10⁻²¹ W/Hz, or −173.98 dBm/Hz; this calculator uses that figure and AN 57-1's round −174 is the same number to the nearest dB. The noise figure of a device is therefore defined against a 290 K source, whatever source it ends up driven from.

Three names describe the same property. The ratio itself, F, is the noise factor; AN 57-1 notes that it "is often called “noise figure”, more often it is called “noise factor” or sometimes “noise figure in linear terms”. Modern usage of “noise figure” usually is reserved for the quantity NF, expressed in dB units", NF = 10 log F (Eq 1-4). The third is the effective input noise temperature, the temperature of a source resistance that would add the same noise into a noiseless device, related to F by AN 57-1's Eq 1-6, T_e = T₀(F − 1). AN 57-1 notes that "Quite often temperature units are used for devices used in satellite receivers", where, as the same page goes on, the antenna's noise can be far below 290 K.

Noise figure is also independent of bandwidth: the noise at the input and the noise added are both proportional to it, so, as AN 57-1 puts it, "the bandwidth in the numerator of (1-2) cancels with that of the denominator". Bandwidth only enters when you turn a noise figure into a noise power, which is what the noise-floor rows of the calculator do.

Noise figure to noise temperature: the conversion table

The two scales are far from linear in each other. From 0.5 to 0.6 dB, T_e rises by 7.6 K; from 10.0 to 10.1 dB, by 67.5 K. The table runs Eq 1-4 and Eq 1-6 over the range where most amplifiers sit.

One row deserves remembering. At F = 2, NF = 3.01 dB, the device adds exactly as much noise as a 290 K source delivers: its T_e is 290 K. A 3 dB figure is the common shorthand, and at 3 dB T_e is 289 K, close enough that the two are used interchangeably.

The Friis formula: the noise figure equation for a cascade

A chain of stages has one noise figure, and AN 57-1 derives it on page 9. The noise from the source is amplified by every gain; the noise stage 1 adds is amplified by every gain after it; and so on down the chain. Referred back to the input, each stage's added noise is divided by the gain ahead of it: F_sys = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁G₂) + …, AN 57-1's Eq 2-3, which it notes "is often called the cascade noise equation". It is also the Friis formula, after the same Harold Friis. For two stages the note names the term that matters: "The quantity (F2-1)/G1 is often called the second stage contribution. One can see that as long as the first stage gain is high, the second stage contribution will be small."

Two things about the equation are easy to get wrong and the calculator handles both. Every quantity in it is a plain ratio, not a decibel figure; Keysight's 5989-5742 says it directly: "The cascaded noise figure is calculated using gain and noise figure in linear terms rather than in decibels (dB)." And the stages are in signal order, from the antenna or source end, because the division is by the gain ahead of each stage, not behind it. The calculator shows each stage's term, its share of the chain's added noise (F_sys − 1), and the noise figure of the chain as far as that stage, which is Eq 2-3 truncated there; the bars in the figure are that running figure.

Noise temperatures cascade the same way, which follows from T_e = T₀(F − 1): T_e,sys = T_e1 + T_e2/G₁ + …. For choosing which of two amplifiers goes first, AN 57-1's glossary (p. 24) defines the noise measure M = (F − 1)/(1 − 1/G), and "the amplifier with the lowest M should be used at the input to achieve the smallest overall noise figure." MT-006's 4 dB, 25 dB LNA has M = 1.517; Keysight's 83017A, 8 dB at the same gain, has 5.326, and goes second. The calculator lists M for each amplifier with gain once there are two of them.

System noise figure behind an LNA: a table

What the Friis formula means in practice is that the first stage sets the noise figure and the rest barely register, as long as the first stage has gain. The table is the noise figure of two stages: an LNA of the gain and noise figure in the row, followed by a second stage (a receiver, a spectrum analyser, an ADC) of the noise figure in the column. The first two rows are illustrative round numbers, not datasheet values; the others are the parts in the worked examples below.

Reading across the first row, a 1 dB LNA with 20 dB of gain keeps the system within 0.30 dB of its own figure behind a 10 dB second stage. Behind 20 dB it gives up 2.52 dB, and behind 30 dB 9.51 dB: a 20 dB noise factor (100) is as large as 20 dB of gain (100), and the second stage's term is then as large as the first stage's whole noise factor. The rule the table encodes is that the first stage's gain should comfortably exceed the next stage's noise factor, in plain ratios. AN 57-1 names the same boundary, "G≤F2" (p. 10), as the point where second-stage errors "can become significant", and the calculator warns when stage 1 is on the wrong side of it.

Keysight's 5989-5742 prints the same calculation for its own preamplifiers as Table 1 (p. 4), "calculated using equation 2", which is Eq 2-3 again, for a second stage of 13, 15, 20 and 30 dB. Five of its rows, with this calculator's result beside each printed figure:

The first four agree to Keysight's 0.1 dB rounding in every column. The 83050A row does not: with the 21 dB of gain printed for it, Eq 2-3 gives 6.78 dB and 10.76 dB for the 20 and 30 dB columns, where the table prints 6.5 and 9.5 dB. The printed row is what 23 dB of gain gives, 6.10, 6.16, 6.51, 9.54 dB. 23 dB is the gain Table 1 lists for the 83051A, which covers the same 2–26.5 GHz band at the same 6 dB noise figure, and the 83050A's 26.5–50 GHz row fits 23 dB as well (at 21 dB its 30 dB column would be 12.5 dB, not the printed 11.8 dB). Either the gain column or the two rows are misprinted; the note does not say which. Of the table's other 59 printed cells, 54 match the equation after rounding and 5 differ by up to 0.2 dB.

Worked example: MT-006's LNA ahead of an ADC

ADI's tutorial MT-006, by Walt Kester, is about the noise figure of ADCs, which comes out high by RF standards: its AD9446 example, a 16-bit converter at 80 MSPS with an 82 dB SNR, works out to 30.1 dB (p. 5). Its Figure 8 (p. 8) is the cascade that answers the problem: "a high-gain (25 dB) low-noise (NF = 4 dB) stage placed in front of a relatively high NF stage (30 dB)—the noise figure of the second stage is typical of high performance ADCs." The calculator's defaults are this chain, with the 40 MHz bandwidth, f_s/2, of MT-006's AD9446 example.

gain      G1 = 10^(25/10)                          = 316.2
factors   F1 = 10^(4/10)                           = 2.512
          F2 = 10^(30/10)                          = 1000
stage 2   (F2 − 1)/G1 = 999 / 316.2                = 3.159
system    F = 2.512 + 3.159                        = 5.671
          NF = 10 log 5.671                        = 7.537 dB
          Te = 290 × (5.671 − 1)                   = 1355 K
floor     kT0·B·F in 40 MHz                        = −90.4 dBm

MT-006 prints "NFT = 10 log105.67 = 7.53dB", "only 3.53 dB higher than the first stage noise figure of 4 dB". Unrounded, the figure is 7.537 dB; MT-006 rounds G₁ to 316 and F to 5.67, and 10 log 5.67 is itself 7.536 dB, so its last digit is truncated rather than rounded. The page's tests hold the calculation to MT-006's figure within 0.01 dB.

The LNA takes the chain from the ADC's 30 dB to 7.54 dB, 22.5 dB better: without it the 40 MHz noise floor referred to the input would be −68.0 dBm, and with it the floor is −90.4 dBm. MT-006's conclusion is the one the table above shows: "The first stage dominates the overall NF", and "It should have the highest gain possible with the lowest NF possible." The share column adds a caution. Of the noise the chain adds, F − 1, the ADC still supplies 68 % and the LNA 32 %, because G₁ = 316 is smaller than F₂ = 1000: this is AN 57-1's "G≤F2" case, and the calculator flags it. More gain ahead of the ADC, or a second amplifier, is what would bring the chain closer to the LNA's own 4 dB.

Worked example: Keysight's preamplifier ahead of a spectrum analyser

Keysight's 5989-5742 works a sensitivity example on page 7: a spectrum analyser with "a noise floor of –110 dBm in a 10 kHz resolution bandwidth and a noise figure of 24 dB", and a preamplifier with "a gain of 36 dB and a noise figure of 8 dB". Enter it in the calculator as two active stages, 36 dB / 8 dB then 0 dB / 24 dB, with B = 0.01 MHz.

alone     analyser: kT0 + 10 log(10 kHz) + 24 dB   = −110.0 dBm
cascade   F = 6.310 + (251.2 − 1)/3981             = 6.372
          NF = 10 log 6.372                        = 8.04 dB
floor     kT0 + 40 dB + 8.04 dB                    = −125.93 dBm
Keysight  8 dB − 2.5 dB − 174 + 40                 = −128.5 dBm

The analyser alone checks out: 24 dB over kT₀ in 10 kHz is −110.0 dBm, Keysight's −110 dBm. With the preamplifier, Keysight takes the system noise figure as "that of the preamplifier less 2.5 dB or 5.5 dB" and prints a sensitivity of −128.5 dBm. The 2.5 dB is not part of the cascade. It is a display effect, stated on page 5: "A spectrum analyzer using log power averaging displays a random noise signal 2.5 dB below its actual value." The Friis cascade of the same two stages is 8.04 dB, the preamplifier's 8 dB plus a 0.04 dB second-stage contribution, and the noise power in 10 kHz is −125.9 dBm. That is the power; the log-averaged trace sits about 2.5 dB lower, near Keysight's figure. Use the calculator's number for a receiver and Keysight's for what the analyser screen shows.

Losses before the LNA: cable, filters and pads

A passive loss is a stage with a gain below one. AN 57-1 says so for passive mixers, "All the equations for noise figure still apply; however, the linear gain values used will be less than one" (p. 16), and adds the consequence: "the second stage noise contribution can be major". Its noise figure is its loss: Keysight's 5989-5742: "If only loss exists in the cascade, then the cascaded noise figure equals the magnitude of the total loss" (p. 3). So the calculator takes a passive stage's noise figure equal to its loss in dB, with no correction for the physical temperature of the loss.

Put a loss in front of the first amplifier and it adds straight to the noise figure, dB for dB: nothing ahead of it has gain to divide it down. Put it behind and it is divided by the amplifier's gain like any other later stage. Take an illustrative chain, a 2 dB cable run to an LNA of 1 dB noise figure and 20 dB gain, then a 10 dB receiver:

• Without the cable: 1.30 dB.
• Cable ahead of the LNA, at the antenna end of the run: 3.30 dB.
• LNA at the antenna, cable behind it: 1.48 dB.

That is 1.82 dB for moving one box to the other end of one cable, and the case for mounting the LNA at the antenna. MT-006's chain with 3 dB of cable ahead of its LNA goes from 7.54 dB to 10.54 dB; with the cable after the LNA it is 9.45 dB, still worse than without it, because the ADC behind has a high noise figure and 25 dB of gain is not very much against it.

The same applies to a filter or an attenuator pad, and it cuts both ways. A 6 dB pad behind the same LNA, there perhaps to improve the match into the receiver or to keep it out of compression, costs 0.87 dB; the same pad ahead of the LNA costs 6.00 dB. When a chain starts with a loss, or has one right behind its first amplifier, the calculator shows what moving it across that amplifier would do.

Noise floor and sensitivity: −174 dBm/Hz + 10 log B + NF

The noise floor of a receiver, referred to its input, is the source's noise plus the chain's added noise over the noise bandwidth: k(T_a + T_e)B. AN 57-1's glossary (p. 24) calls the sum the operating noise temperature, T_op = T_a + T_e. With a 290 K source this is kT₀·B·F, and in decibels it is the familiar sum: −173.98 dBm/Hz, plus 10 log B, plus NF. The first term is the table below; the rest is the calculator. It is derived from kT₀ here, not quoted. The sensitivity is the noise floor plus whatever signal-to-noise ratio the demodulator needs; Keysight's page 7 writes it with 0 dB, "NFsys + kTBB + 10 log(RBW/1)", which is the calculator's default.

B is the noise bandwidth, not the −3 dB bandwidth. MT-006 notes that "the noise bandwidth of a filter is always greater than the 3-dB bandwidth of the filter by a factor which depends upon the sharpness of the cutoff region of the filter" (p. 3), and tabulates 1.57 for one Butterworth pole and 1.11 for two; the Johnson noise calculator applies the same factors.

The source temperature matters when it is not 290 K. AN 57-1 (p. 8): "In satellite receivers the noise level coming from the antenna can be far less, limited by sidelobe radiation and the background sky temperature to values often below 100K. In these situations, a 3 dB change in the receiver noise figure may result in much more than 3 dB signal-to-noise change." With a 50 K source, a 3 dB amplifier costs 8.3 dB of SNR and a 1 dB one 4.0 dB. Enter the source temperature and the calculator reports the SNR lost, 10 log(T_op/T_a), beside the noise figure.

Where the cascade model stops being valid

Gains are available gains into matched ports. AN 57-1 (p. 10): "When an input power of kToB is used in these calculations, it is an available power, the maximum that can be delivered to a matched load." When a stage's input does not match the output of the stage before, "the total gain of a cascaded series of devices does not equal the product of the gains." The note's reassurance is conditional: "If the measurement system has low reflection coefficients and the device has a good output match there will be little error in applying the cascade noise figure equation (2-3) to actual systems." Datasheet gains are usually S₂₁ in a 50 Ω system, which is that condition.

A stage's noise figure depends on what drives it. A noise figure is measured from a specified source impedance, and the glossary (p. 23) is plain: "Characterizing a system by noise figure is meaningful only when the impedance (or its equivalent) of the input termination is specified." A second stage characterised from 50 Ω and driven from a first stage that is not 50 Ω has a different noise figure. AN 57-1 notes the error is divided by the first stage's gain, "Fortunately, the second stage noise contribution is reduced by the first stage gain", except that "When the first stage has low gain (G≤F2), second stage errors can become significant." And for the LNA itself: "The minimum noise figure does not necessarily occur at either the system impedance, Zo, or at the conjugate match impedance that maximizes gain."

Temperature. Noise figure is defined for a 290 K source, and the calculator's noise floor uses the source temperature you enter. A passive stage, though, is entered with its noise figure equal to its loss, as Keysight's note treats losses, and no correction is made for the physical temperature of the cable or filter itself. A cable on a hot roof or a cooled filter is outside what this page models.

Linear stages only. AN 57-1 (p. 8): "The noise figure of a DUT is independent of the signal level so long as the DUT is linear". An amplifier near compression, or an ADC near full scale, is outside the model; so is noise that is not white across the band, such as a mixer's local-oscillator noise, which AN 57-1 (p. 16) notes "can be converted in the mixer to the IF frequency band and become an additional contribution to the system’s noise figure."

Common noise figure mistakes

• Adding noise figures in dB. For MT-006's chain, 4 dB plus the second stage's 30 dB less the first stage's 25 dB gives 9 dB; Eq 2-3 in plain ratios gives 7.54 dB. Adding decibels multiplies the ratios, where the cascade adds them.
• Forgetting the cable ahead of the LNA. Every dB of loss before the first gain is a dB of noise figure: the 2 dB of cable above turned a 1.30 dB chain into 3.30 dB. Enter it as a stage of its own.
• Putting the stages in the wrong order. Eq 2-3 divides each stage by the gain ahead of it; the same stages in reverse give a different answer. Start from the antenna or source end.
• Confusing noise factor with noise figure. F is a ratio, NF is in dB. AN 57-1's glossary (p. 23): "No “dB” symbol implies that the numerical ratio is meant." A noise factor of 2 is 3.01 dB, not 2 dB.
• Using the −3 dB bandwidth for the noise floor. The noise bandwidth is wider, by 57 % for a single pole, per MT-006's table.
• Reading a spectrum analyser's noise floor as the true noise power. With log averaging it displays noise 2.5 dB low (5989-5742 p. 5).
• Assuming a low noise figure is worth the same everywhere. From a cold sky, the SNR lost to a given noise figure is larger than the figure itself; see the source-temperature field.
• Adding gain without counting its cost. Once the first stage's gain is well above the next stage's noise factor, more of it lowers the noise figure little, and it raises the noise and signal the later stages must handle: in Keysight's example (page 8), with the preamplifier fitted "the measurement range is 92.5 dB, 17.5 dB less than without the preamplifier."

Further reading

• Keysight (Agilent) AN 57-1, Fundamentals of RF and Microwave Noise Figure Measurements (5952-8255E) — the definitions (pp. 7–8), the cascade equation (p. 9), gain and mismatch (p. 10), losses (p. 16), and the glossary's noise measure and operating temperature (p. 24).
• Keysight 5989-5742EN, Preamplifiers and System Noise Figure — losses in a cascade (p. 3), Table 1 of system noise figure behind its preamplifiers (p. 4), and the spectrum-analyser sensitivity example (pp. 5–8).
• ADI MT-006, ADC Noise Figure: An Often Misunderstood and Misinterpreted Specification — an ADC's noise figure from its SNR, and the two-stage cascade of Figure 8 (p. 8).
• BIPM, The International System of Units (SI Brochure, 9th edition) — the exact value of the Boltzmann constant behind kT₀.
• Johnson noise calculator — the thermal noise kT that noise figure is measured against, as a voltage across a resistor, and the noise-bandwidth factors.
• dB calculator — converting gains, losses and dBm to and from plain ratios and watts, which is where the Friis formula has to be worked.
• ADC noise floor calculator — the SNR, process gain and FFT floor of a converter, the input to MT-006's ADC noise figure.
• Attenuator pad calculator — the resistor values for a pad, whose loss is also its noise figure.