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Balanced Mixer: LO + RF → IF

Companion to the Balanced Mixers pblog post — multiply LO and RF in the time domain and watch the sum and difference lines appear in the spectrum. Try a classic superhet, then direct conversion where the beat drops into the audio band.

A mixer output is proportional to RF × LO. With sinusoidal inputs at fLO and fRF, the product contains new tones at fLO + fRF and |fLO − fRF|. Real diode rings approximate multiplication by a square-wave LO when the LO is strong enough to switch the diodes.

Controls

fLO = 1000.0 kHz
fRF = 1455.0 kHz
Time domain — LO, RF, and product
Frequency domain — mixing products
Classic AM broadcast-style numbers: RF = 10.455 MHz, LO = 10.000 MHz, difference IF = 455 kHz. The simulator uses the same ratio at a display-friendly scale (LO = 1000 kHz, RF = 1455 kHz). The IF strip in the receiver is built entirely at 455 kHz.

Superheterodyne

RF (tuned)10.455 MHz
LO10.000 MHz
Difference IF455 kHz
Sum product20.455 MHz
The sum at ~20.5 MHz is filtered away before the IF amplifier. Only the 455 kHz difference carries the modulation you care about — unchanged except for the frequency translation.
Spectrum — LO and RF far apart, IF at 455 kHz
IF band zoom (0 – 2 MHz)
In direct conversion, LO sits on top of RF — e.g. LO = 7.100 MHz, RF = 7.101 MHz → IF = 1 kHz (audio). Same mixer as Tab 1, but the time axis is stretched to ~10 ms so the filtered audio beat is visible. The MHz chopping fills the background; the LPF output is the signal you hear.

Controls

fLO = 7.100 MHz
fRF = 7.101 MHz
After low-pass filtering, the IF is a slow beat you can hear in headphones. DC offsets and LO leakage are the practical headaches — not the multiplication itself.
Time domain — audio IF after filtering (long window)
Frequency domain — mixing products

Product of two cosines

For sinusoidal LO and RF,

cos(2π fLO t) · cos(2π fRF t)
= ½ [ cos(2π (fLO + fRF) t) + cos(2π (fLO − fRF) t) ]

The two new frequencies are the sum and difference — your IF products.

Square-wave LO (four-diode ring)

When the LO saturates the diodes, the RF path is switched at the LO rate: $v_\text{IF}(t) \approx s(t)\,v_\text{RF}(t)$ where $s(t)=\pm1$. Multiplying by a square wave adds odd-harmonic terms, so spurious responses appear at $|k f_\text{LO} \pm f_\text{RF}|$ for odd $k$. A double-balanced ring cancels LO and RF feedthrough at the IF port; it does not remove every harmonic product. In the simulator's square-wave mode, the spectrum shows only the mixing products — not LO or RF — mirroring the balanced cancellation.

Superheterodyne vs direct conversion

  • Superhet: |fLO − fRF| is a fixed IF (e.g. 455 kHz). Filters and gain are easy at that frequency.
  • Direct conversion: fLO ≈ fRF, so the difference sits near DC–audio. Saves the IF filter; trades image planning for DC and leakage issues.

Read the full write-up: Balanced Mixers: How LO and RF Become IF

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