Frequency Planning Math

Mathematical reference for the frequency planner. All values are in MHz unless otherwise noted.

See also: Reference Equations for cascade analysis formulas, and Frequency Plan for how to configure this in the UI.

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Heterodyne relationships

A mixer converts a signal from one frequency band to another using a Local Oscillator (LO). The three frequencies — RF, IF, and LO — are always related by:

$$\text{LO}_{\text{high}} = \text{RF} + \text{IF}$$ $$\text{LO}_{\text{low}} = \text{RF} - \text{IF}$$

Given any two of RF, IF, LO the third can be derived.

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High-side vs. low-side injection

The mixing side determines whether the LO is above or below the RF band.

RX downconversion (RF → IF)

Side	Condition	IF
High-side	LO > RF	$\text{IF} = \text{LO} - \text{RF}$
Low-side	LO < RF	$\text{IF} = \text{RF} - \text{LO}$

TX upconversion (IF → RF)

Side	RF output
High-side	$\text{RF} = \text{LO} - \text{IF}$
Low-side	$\text{RF} = \text{LO} + \text{IF}$

Deriving LO from RF and IF

$$\text{LO}_{\text{high}} = \text{RF} + \text{IF} \qquad \text{LO}_{\text{low}} = \text{RF} - \text{IF}$$

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Image frequency (RX only)

For a given RF and LO, the image frequency is the spurious input that produces the same IF output as the desired RF signal:

$$f_{\text{image}} = 2 \cdot \text{LO} - \text{RF}$$

The image frequency is always on the opposite side of the LO from RF. It is displayed in the frequency spectrum diagram but no image rejection ratio is computed — filtering must be verified by the engineer.

Example (high-side):

• RF = 2440 MHz, LO = 2640 MHz, IF = 200 MHz
• Image = 2 × 2640 − 2440 = 2840 MHz

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Harmonic frequencies (TX only)

Harmonics of the output signal are displayed on the spectrum diagram for reference:

$$f_{H_n} = n \cdot f_{\text{out}}, \quad n = 2, 3, \ldots$$

where $f_{\text{out}}$ is the output band center of a given stage. Harmonics are shown as markers only — no harmonic distortion level is computed.

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Multi-stage cascade

For chains with $N$ mixers, the solver maintains an array of $N+1$ inter-stage frequencies $f[0] \ldots f[N]$, where consecutive entries are the input and output of each mixer.

RX (downconversion)

Signal enters at RF. Each mixer reduces the frequency:

$$f[0] = \text{RF}, \quad f[i+1] = \begin{cases} \text{LO}[i] - f[i] & \text{(high-side)} \\ f[i] - \text{LO}[i] & \text{(low-side)} \end{cases}$$

The final value $f[N]$ is the output baseband/IF. Example:

f[0] = 2440 MHz (RF)
  → Mixer 1: LO = 2640 MHz, high-side → f[1] = 200 MHz
      → Mixer 2: LO = 210.7 MHz, high-side → f[2] = 10.7 MHz

TX (upconversion)

Signal enters at baseband. Each mixer increases the frequency:

$$f[0] = \text{baseband}, \quad f[i+1] = \begin{cases} \text{LO}[i] - f[i] & \text{(high-side)} \\ \text{LO}[i] + f[i] & \text{(low-side)} \end{cases}$$

The final value $f[N]$ is the RF output. Example:

f[0] = 10.7 MHz (baseband)
  → Mixer 1: LO = 210.7 MHz, high-side → f[1] = 200 MHz
      → Mixer 2: LO = 2640 MHz, high-side → f[2] = 2440 MHz (RF)

Constraint propagation

The solver runs bidirectionally — it can propagate forward, backward, or derive a missing LO from the two adjacent frequencies:

Known	Derived
$f[i]$, $\text{LO}[i]$	$f[i+1]$
$f[i+1]$, $\text{LO}[i]$	$f[i]$
$f[i]$, $f[i+1]$	$\text{LO}[i]$

This means you only need to enter one value per mixer (LO or the output IF) — the planner fills in the rest.

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Bandwidth propagation

Alongside the center-frequency solver, the planner propagates signal bandwidth through each mixer stage. The behaviour depends on which quantity is held constant:

Constant LO (default)

The LO frequency is fixed; the IF band sweeps with the RF signal. Signal bandwidth passes through the mixer unchanged:

$$BW_{\text{IF}} = BW_{\text{RF}}, \quad BW_{\text{LO}} = 0$$

Constant IF

The IF center is fixed; the LO tracks the RF band. Signal bandwidth appears on the LO instead of the IF output:

$$BW_{\text{LO}} = BW_{\text{RF}}, \quad BW_{\text{IF}} = 0$$

Current behaviour: The planner defaults to Constant LO for all mixer stages. Per-stage mode selection is not yet exposed in the UI.

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