Reference Equations

All formulas used by the analysis engine. No derivations — only the equations applied, their inputs, and references.

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Total gain

$$G_{\text{total}} = G_1 + G_2 + \cdots + G_n \quad \text{(dB)}$$

All stage gains are summed in dB. Passive stages (filters, attenuators) contribute negative gain equal to their insertion loss.

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Power propagation

$$P_{\text{out}} = P_{\text{in}} + G \quad \text{(dBm)}$$

Requires Input Power to be set in the system spec. Without it, absolute power levels are UNKNOWN.

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Cascaded noise figure (Friis formula)

Convert each stage's Noise Figure and Gain from dB to linear:

$$F_i = 10^{NF_i / 10}, \qquad g_i = 10^{G_i / 10}$$

Cascade:

$$F_{\text{cas}} = F_1 + \frac{F_2 - 1}{g_1} + \frac{F_3 - 1}{g_1 g_2} + \cdots + \frac{F_n - 1}{g_1 g_2 \cdots g_{n-1}}$$

Convert back to dB:

$$NF_{\text{cas}} = 10 \log_{10}(F_{\text{cas}}) \quad \text{(dB)}$$

Implication: Stages early in the chain dominate the cascaded NF. A high-NF first stage is almost impossible to recover from, regardless of later stages.

Reference: Friis, H.T. (1944). "Noise Figures of Radio Receivers." Proc. IRE, 32(7), 419–422.

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Cascaded IP3

This example uses "Output-referred" IP3 (OIP3), as opposed to "Input-referred" IP3 (IP3). The relationship between IIP3 and OIP3 is:

$$OIP3_{\text{dBm}} = IIP3_{\text{dBm}} + G_{\text{dB}}$$

Convert each stage's OIP3 and Gain from dB to linear:

$$oip3_i = 10^{OIP3_i / 10}, \qquad g_i = 10^{G_i / 10}$$

Cascade:

$$\frac{1}{OIP3_{\text{total}}}= \frac{1}{OIP3_1} + \frac{1}{OIP3_2 \cdot G_1} + \frac{1}{OIP3_3 \cdot G_1 G_2} + \cdots + \frac{1}{OIP3_N \cdot G_1 G_2 \cdots G_{N-1}}$$

Convert to dBm:

$$OIP3_{\text{dBm}} = 10 \log_{10}(oip3_{mW}) \quad \text{(dBm)}$$

Cascaded IIP3:

$$IIP3_{\text{cas}} = OIP3_{\text{cas}} - G_{\text{total}} \quad \text{(dBm)}$$

Implication: Each stage's nonlinearity contribution is scaled by the gain before it, so later stages dominate the overall OIP3 and the final amplifier typically sets the linearity limit.

Reference: Razavi, B. (2012). RF Microelectronics, 2nd ed. Prentice Hall. Ch. 2.

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Cascaded OIP2

Similar recursive structure as OIP3:

$$\sqrt{\frac{1}{oip2_{\text{cas}}}} = \sqrt{\frac{1}{oip2_{1} \cdot g_2}} + \sqrt{\frac{1}{oip2_2}}$$

Convert to dBm:

$$OIP2_{\text{dBm}} = 10 \log_{10}(oip2_{mW}) \quad \text{(dBm)}$$

Cascaded IIP2:

$$IIP2_{\text{cas}} = OIP2_{\text{cas}} - G_{\text{total}} \quad \text{(dBm)}$$

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Image frequency (heterodyne chains)

When a mixer is present in the chain, the image frequency (an unwanted sideband from the frequency conversion) is computed as:

$$\text{Image} = 2 \times LO - RF$$

This formula is valid for both high-side and low-side LO injection. The result is mathematically equivalent to:

• High-side: $\text{Image} = RF + 2 \times IF$
• Low-side: $\text{Image} = RF - 2 \times IF$

The image location is important for filter design and spur management. If the image falls within the RF band, image rejection filtering becomes critical to prevent interference.

Note: The image frequency is computed directly from LO and RF center frequencies and is independent of the explicit IF value.

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Cascaded OP1dB

Calculations for cascaded P1dB are done using "Output-referred" P1dB. The relationship between "Output-referred" to "Input-referred" is:

$$OP1dB_{\text{dBm}} = IP1dB_{\text{dBm}} + (G_{\text{dB}} - 1)$$

First convert to linear:

$$op1dB = 10^{\frac{OP1dB}{10}}$$ $$g = 10^{\frac{G}{10}}$$

Cascade:

$$\frac{1}{op1dB_{\text{total}}}= \frac{1}{op1dB_1} + \frac{1}{op1dB_2 \cdot g_1} + \frac{1}{op1dB_3 \cdot g_1 g_2} + \cdots + \frac{1}{op1dB_n \cdot g_1 g_2 \cdots g_{n-1}}$$
Convert to dBm: $$OP1dB_{\text{dBm}} = 10 \log_{10}(op1dB_{mW}) \quad \text{(dBm)}$$

Cascaded IP1dB:

$$IP1dB_{\text{cas}} = OP1dB_{\text{cas}} - (G_{\text{dB}} - 1) \quad \text{(dBm)}$$

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Headroom

Headroom refers to the dB margin between a signal level and a specified nonlinearity or limit.

$$\text{P1dB Headroom} = P1dB_{\text{cas}} - P_{\text{out}}$$

Warning triggered when calculated headroom < Headroom Margin (default: 3 dB). Color coding: green ≥ margin, yellow 0 to margin, red < 0.

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Noise spectral density (NSD)

$$NSD_{\text{in}} = 10\log_{10}(kT_{\text{sys}}) + 30\ \text{dB} + NF_{\text{src}} \quad \text{(dBm/Hz)}$$

where:

• $k$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• $T_{\text{sys}}$ = system noise temperature (K), default 290 K
• $NF_{\text{src}}$ = source noise figure (dB), default 0 dB
• $+30\ \text{dB}$ converts dBW/Hz to dBm/Hz

$$NSD_{\text{out}} = NSD_{\text{in}} + G_{\text{total}} + NF_{\text{total}} \quad \text{(dBm/Hz)}$$

where:

• $G_{\text{total}}$ = total cascaded gain (dB)
• $NF_{\text{total}}$ = cascaded noise figure (dB)

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Noise floor

$$\text{Noise Floor} = NSD_{\text{out}} + 10\log_{10}(BW_{\text{RF}}) \quad \text{(dBm)}$$

where $BW_{\text{RF}}$ = RF signal bandwidth (Hz), from RF Bandwidth in the system spec.

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SNR

$$\text{SNR} = P_{\text{out}} - \text{Noise Floor} \quad \text{(dB)}$$

where $P_{\text{out}}$ = estimated output power (dBm).

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Output voltage

$$V_{\text{RMS}} = \sqrt{P_{\text{out,W}} \cdot R} \times 1000 \quad \text{(mV)}$$ $$V_{\text{PP}} = V_{\text{RMS}} \times 2\sqrt{2} \quad \text{(mVpp)}$$

where:

• $P_{\text{out,W}}$ = output power in watts
• $R$ = system impedance (Ω), default 50 Ω

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IMD

$$IMD3 = 2(P_{\text{out}} - OIP3_{\text{cas}}) \quad \text{(dBc)}$$ $$IMD2 = P_{\text{out}} - OIP2_{\text{cas}} \quad \text{(dBc)}$$

where:

• $OIP3_{\text{cas}}$ = cascaded output third-order intercept (dBm)
• $OIP2_{\text{cas}}$ = cascaded output second-order intercept (dBm)

Results are negative — intermodulation products are below the carrier.

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Harmonic estimates

$$H_3 = IMD3 - 9.54 \quad \text{(dBc)}$$ $$H_2 = IMD2 - 6.0 \quad \text{(dBc)}$$

Device-class estimates derived from IMD levels.

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SFDR

$$\text{SFDR} = \frac{2}{3}\left(OIP3_{\text{cas}} - NSD_{\text{in}} - 10\log_{10}(BW_{\text{RF}}) - NF_{\text{total}} - G_{\text{total}}\right) \quad \text{(dB)}$$

where $BW_{\text{RF}}$ = RF signal bandwidth (Hz).

Requires: cascaded OIP3, cascaded NF, total gain, and RF Bandwidth.

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Sensitivity (RX)

$$\text{Sensitivity} = \text{Noise Floor} + \text{SNR}_{\text{req}} \quad \text{(dBm)}$$

where $\text{SNR}_{\text{req}}$ = required SNR (dB), defaults to 0 dB when not set in the system spec.

Requires: RF Bandwidth and cascaded NF.

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Psat margin (TX)

$$\text{Psat margin} = P_{\text{sat}} - P_{\text{out,stage}} \quad \text{(dB)}$$

Shown per stage for blocks with Output Psat set (PA, driver). A positive margin means the stage is operating below saturation.

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Unit conversions used internally

Conversion	Formula
dBm → W	$P_W = 10^{(P_{\text{dBm}} - 30) / 10}$
W → dBm	$P_{\text{dBm}} = 10 \log_{10}(P_W) + 30$
dB → linear	$x = 10^{x_{\text{dB}} / 10}$
linear → dB	$x_{\text{dB}} = 10 \log_{10}(x)$
MHz → Hz	multiply by $10^6$

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References

• Friis, H.T. (1944). "Noise Figures of Radio Receivers." Proc. IRE, 32(7), 419–422.
• Razavi, B. (2012). RF Microelectronics, 2nd ed. Prentice Hall. Ch. 2.
• Pozar, D.M. (2012). Microwave Engineering, 4th ed. Wiley. Ch. 10.