RF chain

The cumulative gain, Gcum_n after n stages, is given by

${\displaystyle Gcum_{n}=Gcum_{n-1}\times G_{n}}$

and Gcum_n(dB) is given by

${\displaystyle Gcum_{n}(dB)=Gcum_{n-1}(dB)+G_{n}(dB)}$

where Gcum_n-1(dB) is the total gain of the first (n-1) stages and G_n(dB) is the gain of the nth stage.

Conversion equations between dBs and linear terms are:

${\displaystyle G=10^{G(dB)/10}=\exp {\big (}0.23026\times G(dB){\big )}}$

and

${\displaystyle G(dB)=10\times \log _{10}G=4.6429\times \ln(G)}$

The cumulative noise factor, after n stages of the overall cascade, Fcum_n is given by

${\displaystyle Fcum_{n}=Fcum_{n-1}+{\frac {F_{n}-1}{Gcum_{n-1}}}}$

where Fcum_n-1 is the noise factor of the first (n-1) stages, F_n is the noise factor of the nth stage, and Gcum_n is the overall gain of n stages.

The cumulative noise figure is then

${\displaystyle NFcum_{n}(dB)=10\times log_{10}(Fcum_{n})=4.6429\times ln(Fcum_{n})}$

• Note 1: the use of an amplifier with high gain for the first stage will ensure that the noise figure degradations by later stages will be small or negligible. This will be best for system sensitivity, see later.
• Note 2: for a passive (lossy) section of the chain, the noise figure of the section equals the loss of that section.[15][16]^:55 So, for example, a 3 dB attenuator has a noise figure of 3 dB.

For spreadsheet purposes, it is convenient to refer the 1 dB compression point[14][17] to the input of the RF chain, i.e. P1cum_n(input),

${\displaystyle P1cum_{n}(input)={\frac {1}{{\frac {1}{P1cum_{n-1}(input)}}+{\frac {Gcum_{n}}{P1_{n}(input)}}}}}$

where P1cum_n-1 is the 1 dB compression point at the input of the first (n-1) stages, P1_n is the 1 dB compression point for the nth stage, referred to its input and Gcum_n is the overall gain including the nth stage. The units are [mW] or [Watt].

• Note: for the best result, i.e. a system tolerant to high level signals, is achieved with a low front end gain. This is in conflict with the need for a low overall noise factor, which requires a high first-stage gain.
• Note 2: The 1 dB compression point is abbreviated as P1dB, iP1dB, or oP1dB. It is referenced to input or output power level measured in [dBm]. Overall system performance can be practically evaluated by the 1 dB compression method.

Related parameters, such as IP3 or IM3 are helpful fictive numbers used to evaluate the system. The device would burn, applying IP3 input level. Accuracy of the measurement with spectrum analyzer is (HP/Agilent specs: +-1.0 dB, and +-0.5 dB custom device). Don't chase fractions of dB. In linear systems, this all results in AGC.

The thermal noise power present at the input of an RF chain,[18]^:44[19]^:435[20]^:229 is a maximum in a resistively matched system, and is equal to kTB, where k is Boltzmann's constant (= 1.38044 × 10⁻²³ J/K), T is the absolute temperature, in kelvins and B is the bandwidth in Hz.

At a temperature of 17 °C (≡ 290 K), kTB = 4.003 × 10⁻¹⁵ W/MHz ≡ -114 dBm for 1 MHz bandwidth.

The thermal noise after n stages of an RF chain, with total gain G_T and noise figure F_T is given by

${\displaystyle Ncum_{n}=kTB\times Fcum_{n}\times Gcum_{n}}$

where k = Boltzmann's constant, T is the temperature in kelvins and B is the bandwidth in hertz, or

${\displaystyle Ncum_{n}(dBm/Mhz)=-114+Fcum_{n}(dB)+Gcum_{n}(dB)}$

where Ncum_n(dBm) is the total noise power in dBm per 1 MHz of bandwidth,

In receivers, the cumulative gain is set to ensure that the output noise power of the chain at an appropriate level for the signal processing stages that follow. For example, the noise level at the input to an analog-to-digital converter (A/D) must not be at too low a level, otherwise the noise (and any signals within it) is not properly characterized (see the section on A/Ds, later). On the other hand, too high a level results in the loss of dynamic range.

Sometimes performance at high signal levels is defined by means of the “second-order intercept point (I2)” and the “third-order intercept point (I3)”, rather than by the 1 dB compression point.[14] These are notional signal levels which occur in two-signal testing and correspond to the theoretical points where second and third order inter-modulation products achieve the same power level as the output signal.[1]^:685[3]^:91 The figure illustrates the situation.

Compression and IM Products

In practice, the intercept levels are never achieved because an amplifier has gone into limiting before they are reached, but they are useful theoretical points from which to predict intercept levels at lower input powers. In dB terms, they decrease at twice the rate (IP2) and three times the rate (IP3) of the fundamental signals.

When products, stage to stage, add incoherently, the cumulative results for these products are derived by similar equations to that for the 1 dB compression point.

${\displaystyle I2cum_{n}(input)={\frac {1}{{\frac {1}{I2cum_{n-1}(input)}}+{\frac {Gcum_{n}}{I2_{n}(input)}}}}}$

where I2cum_n-1 is the second order intercept point at the input of the first (n-1) stages, I2_n is the third order intercept point for the nth stage, referred to its input and Gcum_n is the overall gain including the nth stage.

Similarly,

${\displaystyle I3cum_{n}(input)={\frac {1}{{\frac {1}{I3cum_{n-1}(input)}}+{\frac {Gcum_{n}}{I3_{n}(input)}}}}}$

where I3cum_n-1 is the third order intercept point at the input of the first (n-1) stages, I3_n is the third order intercept point for the nth stage, referred to its input.

The cumulative intercept points are useful when determining the “spurious free dynamic range” [16]^:519 of a system.

There is an approximate relationship between the third order intercept level and the 1 dB compression level which is[21]^:59 [20]^:35

${\displaystyle IP3(dB)\approx P1(dB)+11(dB)}$

Although only an approximation, the relationship is found to apply to a large number of amplifiers.[17]

In the spread sheet, the total frequency band of interest B(Hz) is divided into M sub-bands (spreadsheet columns) of B/M (Hz) each, and for each sub-band (m = 1 to M) the thermal noise power is derived, as described above. In practice, these results will differ slightly, from column to column, if the system has gain ripple.

The signal-to-noise ratio (S:N) is the peak signal power of the pulse (Psig) divided by the total noise power (Pnoise) from the M frequency bins, i.e.

${\displaystyle {\frac {S}{N}}={\frac {P_{sig}}{\sum _{m=1}^{M}P_{noise}(m)}}}$

This is the S:N ratio at RF frequencies. It can be related to the video S:N ratio as shown next.

For spreadsheet purposes it can useful to find the RF signal to noise ratio which corresponds to a desired video signal to noise figure after demodulation or detection. As an RF chain usually has sufficient gain for any noise contribution from the detector diode to be ignored, the video S:N can be shown to be[21]^:115

${\displaystyle {\Big (}{\frac {S}{N}}{\Big )}_{vid}={\frac {G^{2}.P_{S}^{2}}{4.G^{2}.P_{S}.kTF'B_{V}+2(kTF'G)^{2}.(B_{V}.B_{R}-B_{v}^{2}/2)}}}$

where

• P_S =input RF signal power;
• 8B_V and B_R are the video and RF bandwidths;
• F' = F -1/G where G is the chain gain and F the effective noise figure;
• k = Boltzmann's constant; and
• T = the ambient temperature

[If there is significant gain variation across the band, then it can be divided into M sub-bands and results summed for these sub-bands, as described earlier.]

From the above equation, as the noise power in the RF band is P_N = kTB_RF’, a relationship between RF and Video S:N ratios can be found.

${\displaystyle {\Big (}{\frac {S}{N}}{\Big )}_{vid}={\frac {{\Big (}{\frac {S}{N}}{\Big )}_{rf}^{2}}{4.{\Big (}{\frac {S}{N}}{\Big )}_{rf}.{\frac {B_{V}}{B_{R}}}+2.{\frac {B_{V}}{B_{R}}}-{\Big (}{\frac {B_{V}}{B_{R}}}{\Big )}^{2}}}}$

(This result can be found elsewhere[22]^:188).

Inverting the relationship gives the RF signal-to-noise ratio required to achieve a given video S:N ratio:

${\displaystyle {\Big (}{\frac {S}{N}}{\Big )}_{rf}={\frac {B_{V}}{B_{R}}}.{\Bigg [}2.{\Big (}{\frac {S}{N}}{\Big )}_{vid}\pm {\sqrt {4.{\Big (}{\frac {S}{N}}{\Big )}_{vid}^{2}+{\Big (}{\frac {S}{N}}{\Big )}_{vid}.{\Big (}2.{\frac {B_{R}}{B_{V}}}-1{\Big )}}}{\Bigg ]}}$

Signal sensitivity is important for receiving systems and refers to the minimum signal level at the input that is necessary to give a signal that can be resolved reliably by the detection process at the end of the RF chain. This parameter is less important in the case of repeaters and transmitter drivers where signal levels tend to be higher and other concerns such as stage overload and spurious signal generation tend to be more relevant.

Determining a value for system sensitivity can be difficult and depends on many things, including the method of detection, the signal coding method, the bandwidth of the RF channel, and whether or not digital processing is involved. Two important parameters used in assessing sensitivity performance of a system are[23]^:2.16 [15]^:204 the "Probability of Detection" and the "False Alarm Rate".

Statistical methods are often used in the decision process (see Tsui[24]^:20 and Skolnik[25]^:16).

Tangential Sensitivity

Tangential sensitivity, (TSS), defines that input power which results in a video signal to noise ratio of approximately 8 dB from the detector.[24]^:16 The thumbnail shows an example of a typical detected pulse at the TSS limit, with the pulse + noise sitting at a level just clear of the noise floor. The TSS level is too low a value for reliable pulse detection in a practical scenario, but it can be determined with sufficient accuracy in bench tests on a receiver to give a quick guide figure for system performance.

In a wideband receiver, with a square-law detector, the TSS value at the chain input terminals is given by,[24]^:18

${\displaystyle TSS(dBm)=114+10.logF_{T}+10.log{\Big (}6.31B_{V}=2.5{\sqrt {2.B_{V}B_{R}-B_{V}^{2}}}{\Big )}}$

From this, the S:N of the RF signal, at the input to the detector can be obtained when the video output is at TSS.

${\displaystyle {\Big (}{\frac {S}{N}}{\Big )}_{R}(dB)=10.log{\Bigg [}{\frac {B_{V}}{B_{R}}}{\Bigg (}6.31+2.5{\sqrt {2.{\frac {B_{R}}{B_{V}}}-1}}{\Bigg )}{\Bigg ]}}$

This equation shows that the S:N at RF is typically less than unity, in wideband systems, when the video output is at TSS. For example, if B_R/B_V = 500 then the equation gives (S:N)_R = 0.17 (≈ -7.7 dB). (Note: a similar result is obtained by using the equation relating RF and video S:N ratios, given in the previous section[22]^:190).

The thumbnail shows the simulated video output (at TSS) corresponding to an RF pulse in wideband noise with S:N = 0.17 and a bandwidth ratio of 500.

The sensitivity of a system may be taken as the "minimum detectable signal". This is that level of signal that exceeds a threshold value by a suitable margin (If the level is set too low, noise spikes will exceed it too frequently and if the signal+noise does not exceed it by a sufficient margin then it may fall below the threshold giving pulse termination prematurely. So, in determining the minimum detectable signal, it is necessary to choose the "false alarm rate" and "probability of detection" values appropriate to the system requirement. To aid the designer, graphs are available, to help determine the necessary S:N ratio at the detector.[24]^:30[25]^:28[26]^:2.19[27]^:21[15]^:357

In the case of pulse detection of a signal in noise, following the detector in a wideband receiver, where the RF bandwidth greatly exceeds the video bandwidth, a guideline figure for reliable performance a S:N (at video) is 16 to 18 dB.[21]^:87 This is a useful figure for use in spreadsheets, and it corresponds to a probability of detection of over 99% for a Swerling 1 target[28][29]

(Although lower values of S:N can give acceptable "Probability of Detection" and "False Alarm Rate" figures, the measurement of pulse lengths become less reliable because noise spikes on pulses may extend below the chosen threshold level).

Pulses with 18 & 15dB S:N

As examples, thumbnails show simulated examples of a detected pulse, in noise, where S:N = 18 dB and 15 dB. As can be seen, if the S:N falls to 15 dB or lower, it becomes difficult to set a threshold level for pulse detection, that is clear of the noise floor, and yet does not result in early termination.

The video S:N ratio can be related to the RF S:N ratio, as shown earlier.

In scenarios, such as radar pulse detection, integration over several pulses may occur and a lower value of S:N then becomes acceptable.[25]^:30 In general, system sensitivity and pulse detection theory are specialized topics [20]^:12 and often involve statistical procedures not easily adapted for spreadsheets.

If a transmission line is mismatched at both ends, multiple-reflected signals can be present on the line, resulting in ripple on the frequency response, as seen at the load.

Where only first time round echoes are considered (i.e. multiple reflections are ignored), the output response is given by

${\displaystyle V_{out}=V_{1}(1+\alpha ^{2}.\rho _{1}.\rho _{2}.e^{j.2*\pi .f.2T_{d}})}$

Response of a mismatched cable

Where

• α is the loss for a single pass through the cable,
• ρ₁ and ρ₂ are the voltage reflection coefficients of the terminations,
• f is the frequency,
• T_d is the (single pass) propagation delay of the cable

A typical plot is shown in the thumbnail.

This response has a ripple component with a peak-to-peak value ΔA, given by

${\displaystyle \Delta A=2.{\frac {1+\rho _{1}.\rho _{2}}{1-\rho _{1}.\rho _{2}}}}$

The frequency difference from peak-to-peak (or trough-to-trough) of the ripple is given by ΔΩ where

${\displaystyle \Delta \Omega ={\frac {\pi }{T_{d}}}}$

An RF chain may contain many inter-stage links of various lengths. The overall result is obtained using

${\displaystyle {\text{Overall Result}}=\prod _{n=1}^{N}Vout_{n}}$

This can give an overall response which is far from flat. As an example, a random collection of 25 cascaded (but separated) links give the result shown. Here, a random selection of path delays are assumed, with α taken as unity and ρ₁ and ρ₂ taking the typical value 0.15 (a return loss ≈ 16 dB), for the frequency range 10 to 20 GHz

The Response of Multiple Mismatches

For this example, calibration at 50 MHz intervals would be advisable, in order to characterize this response.

The ripple amplitude would be reduced if the mismatches ρ₁ and ρ₂ were improved but, especially if the lengths of the interconnecting links were made shorter. An RF chain, made up of surface mounted components, interconnected by stripline,[4]^:168 which can be made physically small, may achieve less than 0.5 dB ripple. The use of integrated circuits would give lower ripple still, (see, for example Monolithic microwave integrated circuits).

Detector diodes for RF and Microwaves may be point contact diodes, Schottky diodes, Gallium Arsenide or p-n junction devices.[44] Of these, Schottky diodes and junction diodes require biassing for best results. Also, silicon junction diodes perform less well at high frequencies. A typical detector diode has a TSS of -45 to -50 dBm [24]^:136[45][46] and peak pulse powers of 20dBm, although better figures are possible[47]).

At low powers, diodes have a square-law characteristic, i.e. the output voltage is proportional to the input power, but at higher powers (above about -15dBm) the device becomes linear, with the output voltage proportional to the input voltage.

Square law detectors can give detectable signals at video, in wideband systems, even when the RF S:N is less than unity. For example, using the RF-to-Video relationships given earlier, for a system which has a bandwidth of 6 GHz, and an RF S:N value of 0.185 (-7 dB), the video S:N (i.e. TSS) will be 6.31 (8 dB). (Tsui's equations give an RF S:N value of 0.171 for this example).

DLVAs[48][21]^:72 have been commonly found in direction finding systems, using multiple channels, squinted antennas and amplitude comparison methods.[21]^:155[49] They are also useful for compressing the dynamic range of incoming signals of receivers, prior to digitising. They cover frequency ranges such as 2 – 6 GHz and 6 – 18 GHz. There are also wideband devices available which cover the range 2 – 18 GHz.

A simple DLVA contains a broadband diode detector followed by an amplifier with a logarithmic characteristic and has an input power range of, typically, -45dBm to 0dBm,[50][51][52] which may be increased to -45 to +15dBm in an extended-range DLVA. Two devices, together with an amplifier, can be combined to give an effective range of -65dBm to +15dBm.

In a successive-detection DLVA, which includes a low noise amplifier, the power range may to be, typically -65dBm to +10dBm[53][51]

IFMs can provide a frequency measurement of a single pulse.[21]^:126:140 They incorporate a set of delay-line frequency discriminators, with delay lengths increasing in a binary or other sequence.[54][55] They usually incorporate some gain of their own. The discriminator with the longest delay line establishes the frequency measurement accuracy and resolution, the shortest delay line correlator defines the unambiguous bandwidth of the DFD and the remaining correlators serve to resolve ambiguities.[56] Usually, there is an input, limiting amplifier present in the IFM. This boosts the received signal to a constant level for processing by the correlators, making the frequency-data-decoding task of the frequency processor easier, and to emphasise the “capture effect” when simultaneous signals are present. Normally the RF amplifier will produce a minimum of 10 dB limiting at the lowest specified signal input level. If the RF S:N ratio is too low, the output of the longest delay line correlator (which sets the frequency resolution of the IFM) will become degraded and noisy. At high S:N ratios (+10dBm), the measured frequency accuracy approaches the correlator-limited rms error, but at approximately -3dBm SNR, ambiguity errors appear, causing large measurement inaccuracies.[57]

The lowest input power level of a typical DDU is about -75dBm,[57] and with a receiver noise figure of 10 dB, it gives a frequency accuracy of approximately 1 MHz[55] They have dynamic ranges of 65 to 75 dB and cover frequency bands such as 2 – 6 GHz, 6–18 GHz and some wideband devices cover 2 – 18 GHz.[58][59][55]

With the advent of digital techniques, analogous processes to those of an analog system have been realized.[60][61]

An Analog-to-digital converter,[62][63] located at the end of the RF chain, provides digital signals for further signal processing. As the A/D operates with sampled signals, it is necessary for the Nyquist–Shannon sampling theorem to be satisfied, if data is not to be lost. As shown earlier, a low-amplitude RF pulse immersed in wideband noise, can be detected by a square-law diode detector. Similarly spread spectrum signals can be recovered from below the noise floor by compression. Consequently, to ensure no loss of data, the chain gain should be high enough to ensure that thermal noise will activate the A/D adequately, so that any signals present within the noise, can be recovered correctly by the detection or compression process. Typically, the rms noise voltage present the input to the A/D should be one or two bits of the A/D range, but no lower. On the other hand, having excessive chain gain so that the noise floor is unnecessarily high, will result in the loss of dynamic range.

Consider, as an example, a chirp signal with time-bandwidth product of 200 and of amplitude of ½ LSB which is embedded in noise with an rms voltage of 1 LSB, present at the input to an A/D. he digitized, quantised output, relative to the mean value, is similar to the example in the left-hand figure below. After compression in the signal processor, a high amplitude pulse, whose magnitude is well above the noise is obtained, as shown in the right-hand figure.

This example happens to show, unintentionally, the benefits of dither[64][65][66] which is used to improve the linearity and dynamic range of an A/D. In the case of the signal considered here, if there was no noise present, but just the signal alone, its amplitude would be insufficient to operate the A/D.