// RF Theory › Mixers
Mixer Spurious Responses
Every mixing product a real mixer can produce — the general formula, each named spur type explained from first principles, worked frequency examples, an interactive spur table generator, and suppression techniques for each. Essential knowledge for receiver and transmitter design.
// Root Cause
Why Mixer Spurs Exist
An ideal mixer would multiply exactly two signals and produce only two output products: the sum and difference of the RF and LO frequencies. A real mixer is implemented with nonlinear devices — diodes, transistors, or FET switches — and nonlinear devices produce an infinite series of harmonic and intermodulation terms, not just the two desired products. Every one of these unintended products is a spurious response, or spur.
The nonlinearity of a diode or transistor can be expanded as a Taylor series in the input voltage. Each term of that series produces a distinct family of output products:
Nonlinear Device Taylor Series
i(v) = a₀ + a₁v + a₂v² + a₃v³ + a₄v⁴ + ...Input: v = A·cos(ω_RF·t) + B·cos(ω_LO·t)
a₁v → fundamental RF and LO terms (no mixing, just amplification)
a₂v² → 2nd-order: DC, 2f_RF, 2f_LO, f_RF+f_LO, |f_RF−f_LO| (wanted IF!)
a₃v³ → 3rd-order: f_RF, f_LO, 3f_RF, 3f_LO, 2f_RF±f_LO, 2f_LO±f_RF, IM3 products
a₄v⁴ → 4th-order: DC, harmonics, and more sum/difference products
... and so on for every order n
The second-order term (a₂v²) produces the wanted IF = |f_RF − f_LO|. But every higher-order term adds more products, many of which fall at frequencies that can mix down to the IF band or directly corrupt the signal. These are the spurs.
Key insight: Spurs are not a design flaw — they are a mathematical inevitability of using any nonlinear device for mixing. The designer's job is to understand which spurs fall in the IF band for a given set of frequencies, quantify their power levels, and suppress them to an acceptable level through topology choice, filtering, and frequency planning.
// Foundation
The General Spur Formula
Every spurious output frequency a mixer can produce is described by a single universal formula involving integer harmonics of the RF and LO inputs:
Universal Spur Frequency Formula
f_spur = | m × f_RF ± n × f_LO |where m, n = 0, 1, 2, 3, ... (non-negative integers)
Spur order = |m| + |n| (total harmonic order)
Wanted product: m=1, n=1 → IF = |f_RF − f_LO| (order 2)
Number of products up to order N ≈ N² (grows rapidly with order)
Understanding Spur Order
The order of a spur is |m| + |n| — the sum of the harmonic numbers involved. It determines two critical things: how easily the spur can be generated, and how rapidly its power changes with input level.
Spur Power vs Order — Why Higher Order = Weaker
Spur Power vs Input Power
P_spur(m,n) ∝ P_RF^m × P_LO^nIn logarithmic terms (dBm):
P_spur ≈ m × P_RF + n × P_LO + K(m,n)
K(m,n) = a constant determined by the mixer topology and device characteristics
Practical consequence:
Order 2 (1,1): P_spur grows 1 dB per 1 dB input (linear — this is the wanted IF)
Order 3 (2,1) or (1,2): P_spur grows 2 dB per 1 dB RF increase — the half-IF and image spurs
Order 4 (2,2): P_spur grows 2 dB per 1 dB each — smaller but still present
Order 5 (2,3) or (3,2): 3 dB per 1 dB — much smaller at typical operating levels
This power-law relationship is why higher-order spurs are generally less problematic at small RF input levels — but at high input levels (near compression), the relative power difference narrows and previously negligible high-order spurs can become significant. This is one reason why strong-signal environments (co-located transmitters, crowded spectrum) are so difficult to design for.
// Complete Spur Catalogue
Every Named Spur Type
While the general formula covers all spurs mathematically, RF engineers have given specific names to the most practically important ones. Each named spur has a characteristic frequency, mechanism, power level, and suppression strategy.
1. Image Frequency Spur
The image is the most fundamental spur — it arises directly from the wanted mixing product (order 2) because mixing is symmetric: two input frequencies, one on each side of the LO, both produce output at the same IF frequency.
Image Frequency
Wanted signal: f_RF = f_LO + f_IF (assuming high-side injection f_RF > f_LO)Image frequency: f_image = f_LO − f_IF = 2·f_LO − f_RF
Image is always 2×f_IF away from f_RF
Both f_RF and f_image produce IF = |f − f_LO| = f_IF
→ They are INDISTINGUISHABLE after downconversion
Example: f_LO=1000 MHz, f_IF=100 MHz
Wanted: f_RF = 1100 MHz Image: f_image = 900 MHz
Both produce exactly 100 MHz at the IF port
Why it's dangerous: A strong interferer at f_image produces exactly the same IF frequency as the wanted signal. No IF filter can remove it — it arrives already downconverted. The only way to suppress the image is BEFORE the mixer: a preselector filter, a higher IF (moves the image further away), or an image-reject mixer architecture.
Image Rejection (IR)
IR = preselector attenuation at f_image (dB)Requirements: FM broadcast → 60 dB; cellular → 50–70 dB; military EW → 80–100 dB
Image reject mixer (Hartley/Weaver): achieves 30–50 dB IR from hardware alone
Digital IQ correction: extends to 50–70 dB
2. Half-IF Spur
The half-IF spur is the most insidious mixer spur because it cannot be moved by changing the IF frequency — it always sits at exactly f_LO + f_IF/2, regardless of the IF chosen.
Half-IF Spur — Mechanism
Interferer at: f_int = f_LO + f_IF/2Step 1: Mixer generates 2nd harmonic of f_int: 2·f_int = 2·f_LO + f_IF
Step 2: Mixer also generates 2nd harmonic of LO: 2·f_LO
Step 3: Mixing: 2·f_int − 2·f_LO = f_IF ← lands EXACTLY in IF band
Spur order: m=2, n=2 → 4th order total
Power: P_spur ∝ P_int² × P_LO² → grows as 2 dB per dB of interferer increase
Why it matters: The interferer is only f_IF/2 above the LO — it sits very close to the wanted signal (which is f_IF above the LO). For a narrow-band IF of 100 kHz (as in GSM), the half-IF interferer sits only 50 kHz above the LO — well inside the RF bandpass filter's passband and impossible to reject with a preselect filter. For a wide-band IF of 100 MHz (as in a dual-conversion superhet), the interferer is 50 MHz above the LO — typically still inside the preselector.
Suppression: High IIP2 in the mixer (double-balanced structures cancel even-order terms), balanced LO drive (removes even LO harmonics), and if possible a higher IF frequency (moves the interferer further outside the preselector passband).
Historical significance: The half-IF spur was the defining receiver design challenge for 1990s GSM chipsets (IF = 270 kHz, half-IF interferer at 135 kHz from LO). It drove intensive research into high-IIP2 double-balanced mixers and is the primary reason why commercial GSM receivers use double-balanced Gilbert cell mixers exclusively — single-balanced mixers have insufficient even-order cancellation.
3. IF Feedthrough (DC Offset)
IF Feedthrough
m=0, n=0 → constant DC term at IF outputm=1, n=0 → RF signal appears directly at IF (no downconversion)
m=0, n=1 → LO signal appears directly at IF
In direct-conversion (f_IF=0):
LO self-mixing: LO leaks to RF port → reflects → mixes with LO → DC at IF
Mechanism: f_LO − f_LO = 0 Hz → DC offset at baseband
Effect: saturates baseband amplifiers, corrupts ADC dynamic range
In a superheterodyne receiver (f_IF ≠ 0), IF feedthrough is less damaging because the DC term is outside the IF passband. In direct-conversion receivers, DC offset is the dominant problem: LO leakage mixes with itself (LO self-mixing) and with reflected versions of itself (from the antenna) to produce a time-varying DC offset that can be tens of millivolts — far larger than the microvolt-level signal the baseband amplifier is trying to amplify.
Suppression: AC coupling (removes DC but cuts low frequencies), adaptive DC cancellation (a DAC continuously subtracts the measured DC offset), or using a non-zero IF (low-IF architecture avoids zero but not entirely).
4. LO Harmonic Spurs
LO Harmonic Spur Family (m=1, n=2,3,4...)
f_spur = | f_RF − n·f_LO | for n = 2, 3, 4...n=2: f_spur = |f_RF − 2·f_LO| → IF = f_RF − 2·f_LO
n=3: f_spur = |f_RF − 3·f_LO| → IF = f_RF − 3·f_LO
Example: f_RF=900 MHz, f_LO=430 MHz (f_IF=40 MHz target)
Wanted (n=1): |900 − 430| = 470 MHz (wrong — this isn't 40 MHz)
For IF=40 MHz: need f_LO = 860 MHz (normal) or 860/2=430 MHz with n=2 (subharmonic!)
Subharmonic mixer deliberately exploits n=2: |900 − 2×430| = |900−860| = 40 MHz ✓
LO harmonic spurs arise because a real LO signal contains harmonics — the LO is not a pure sinusoid. A square-wave LO (used in CMOS passive mixers for sharp switching) contains harmonics at 3f_LO, 5f_LO, 7f_LO (all odd harmonics for a 50% duty-cycle square wave). Each LO harmonic n·f_LO can mix with the RF to produce an IF output at f_RF − n·f_LO. If this frequency falls within the IF filter passband, it is a spurious response.
In a double-balanced mixer: Even LO harmonics (n=2,4,6...) cancel in the balanced structure. Only odd harmonics (n=1,3,5...) produce responses. The most dangerous survivor is (1,3): f_spur = |f_RF − 3·f_LO|.
5. RF Harmonic Spurs
RF Harmonic Spur Family (m=2,3..., n=1)
f_spur = | m·f_RF − f_LO | for m = 2, 3, 4...m=2: f_spur = |2·f_RF − f_LO| → can fall near IF if 2·f_RF ≈ f_LO + f_IF
m=3: f_spur = |3·f_RF − f_LO|
Dangerous condition for m=2:
2·f_RF − f_LO = f_IF → f_RF = (f_LO + f_IF)/2 = (f_IF/2 + f_LO/2)
→ Interferer at half the wanted frequency can produce IF output!
This is a subharmonic response — an interferer at f_RF/2 can create a spur at f_IF
RF harmonic spurs arise because the mixer's nonlinearity generates harmonics of the RF input itself. Even a small signal at f_RF gets squared, cubed, and raised to higher powers by the nonlinear device, producing 2f_RF, 3f_RF and higher harmonics inside the mixer. These harmonics then mix with the LO and its harmonics to produce a dense forest of output products.
In a double-balanced mixer: Even RF harmonics (m=2,4...) cancel. Odd survive. The most dangerous are (3,1) and (3,2).
6. Close-In IM3 Spurs (Two-Tone Intermodulation)
Third-Order Intermodulation Products
Two RF tones: f_A and f_B (both near f_RF)IM3 products: 2f_A − f_B and 2f_B − f_A
After downconversion by LO:
IF_IM3 = |(2f_A − f_B) − f_LO| = |2(f_A−f_LO) − (f_B−f_LO)| = |2·IF_A − IF_B|
Key property: IM3 products fall VERY CLOSE to the wanted IF signals
If f_A and f_B are adjacent channels spaced Δf apart:
IM3 at 2f_A−f_B is INSIDE the IF band, only Δf from f_A
IM3 power: P_IM3 = 3·P_in − 2·IIP3 (all dBm)
IM3/fundamental ratio: IM3 is 2×(IIP3−P_in) dBc below fundamental
IM3 products are the dominant linearity problem in wideband receivers. Unlike the image or half-IF spurs (which are caused by specific interferers at specific frequencies), IM3 products can be created by ANY two interferers in the RF band — and they fall within the IF passband right on top of the wanted signal. No amount of filtering can remove them after the mixer.
IM3 — Worked Example
System: f_LO = 900 MHz, f_IF = 100 MHz, f_RF_wanted = 1000 MHz
Two interferers: f_A = 1005 MHz, f_B = 1010 MHz (5 MHz apart, in adjacent channels)
IM3 products in RF domain: 2×1005−1010 = 1000 MHz and 2×1010−1005 = 1015 MHz
The IM3 at 1000 MHz is EXACTLY the wanted signal frequency!
After downconversion: IF_IM3 = 1000−900 = 100 MHz = f_IF
This IM3 product falls right on top of the wanted channel at 100 MHz IF.
Mixer IIP3 = +10 dBm, interferer power = −20 dBm:
IM3 power = 3×(−20) − 2×10 = −60 − 20 = −80 dBm
If wanted signal is −90 dBm, IM3 is only 10 dB below it — severe degradation of SNR.
Two interferers: f_A = 1005 MHz, f_B = 1010 MHz (5 MHz apart, in adjacent channels)
IM3 products in RF domain: 2×1005−1010 = 1000 MHz and 2×1010−1005 = 1015 MHz
The IM3 at 1000 MHz is EXACTLY the wanted signal frequency!
After downconversion: IF_IM3 = 1000−900 = 100 MHz = f_IF
This IM3 product falls right on top of the wanted channel at 100 MHz IF.
Mixer IIP3 = +10 dBm, interferer power = −20 dBm:
IM3 power = 3×(−20) − 2×10 = −60 − 20 = −80 dBm
If wanted signal is −90 dBm, IM3 is only 10 dB below it — severe degradation of SNR.
7. IM2 / Even-Order Intermodulation
Second-Order Intermodulation
Two RF tones: f_A and f_BIM2 products: f_A + f_B (sum) and |f_A − f_B| (difference)
The difference IM2: |f_A − f_B| is at a very low frequency (baseband)
In direct-conversion (f_IF = 0):
|f_A − f_B| falls directly IN the baseband signal band → catastrophic
Cannot be filtered — the IM2 spur IS IN the signal band
IM2 power: P_IM2 = 2·P_in − IIP2 (dBm)
IIP2 requirement (LTE direct-conversion): > +50 dBm
IM2 is the dominant problem in direct-conversion (zero-IF) receivers. In a superheterodyne receiver, the IM2 difference product |f_A − f_B| falls near DC and is well outside the IF passband (which is centred at f_IF). But in a direct-conversion receiver where f_IF = 0, the IF band IS the baseband, and IM2 falls directly in-band.
Suppression: Double-balanced mixers inherently cancel even-order products (including IM2) when perfectly balanced. Any imbalance — in diode characteristics, layout symmetry, or LO drive balance — breaks the cancellation. This is why direct-conversion chipsets require factory IIP2 calibration: trimming the LO balance or mixer bias to restore the even-order cancellation. Typical post-calibration IIP2 exceeds +60 dBm.
8. Third-Order Cross-Products (m=2,n=1 and m=1,n=2)
Third-Order RF×LO Cross Products
(2,1): f_spur = |2·f_RF − f_LO| — RF 2nd harmonic mixes with LO fundamental(1,2): f_spur = |f_RF − 2·f_LO| — RF fundamental mixes with LO 2nd harmonic
Total spur order = 3 for both
Example (900/800/100 MHz system):
(2,1): |2×900 − 800| = |1800−800| = 1000 MHz → not at IF
(1,2): |900 − 2×800| = |900−1600| = 700 MHz → not at IF
When (2,1) IS at IF:
|2·f_RF − f_LO| = f_IF → f_RF = (f_LO + f_IF)/2
This means an interferer at HALF the wanted frequency creates a spur at f_IF
The (2,1) spur is the half-IF spur's close cousin. While the half-IF spur is generated by a specific relationship between an interferer and the IF bandwidth, the (2,1) product is a more general harmonic mixing term. When f_RF is tuned such that 2·f_RF − f_LO = f_IF, a signal at f_RF creates a spurious response at the IF — even though f_RF is not the intended receive frequency.
9. LO Self-Mix (m=0, n=2) and Blocker Self-Mix
LO Self-Mix and AM-to-DC
LO self-mix: LO leaks to RF port → mixes with itselff_spur = |f_LO − f_LO| = 0 → DC at IF (critical in direct-conversion)
Blocker self-mix (AM-to-IM2):
A single AM-modulated blocker at f_B mixes with itself:
|f_B − f_B| = 0 → the AM envelope appears at DC
The AM envelope = the modulation bandwidth → falls IN the IF band of zero-IF receiver
Two-tone version:
Two close-in blockers f_B1, f_B2: |f_B1−f_B2| appears at low frequency in IF band
The LO self-mix is the primary source of DC offset in direct-conversion receivers. Every path by which the LO leaks to the RF port — through the mixer's finite LO-to-RF isolation, through substrate coupling on the IC, through common ground impedance — results in a DC component at the IF output that drifts with temperature and time.
// Quick Reference
Named Spurs — At a Glance
Image Frequency
Order 2 · (m=1,n=1)
f_image = 2·f_LO − f_RF
Danger: Same IF as wanted signal. Cannot be removed post-mixer.
Suppression: Preselector filter, higher IF, image-reject mixer.
Suppression: Preselector filter, higher IF, image-reject mixer.
Half-IF Spur
Order 4 · (m=2,n=2)
f_int = f_LO + f_IF/2
Danger: Cannot be moved by IF selection. Grows 2 dB/dB.
Suppression: High IIP2, balanced LO, double-balanced mixer.
Suppression: High IIP2, balanced LO, double-balanced mixer.
IM3 Close-In
Order 3 · (m=2,n=1 via two tones)
2f_A − f_B → IF
Danger: Falls in IF band, indistinguishable from signal.
Suppression: High IIP3, reduce LNA gain, balanced mixer.
Suppression: High IIP3, reduce LNA gain, balanced mixer.
IM2 / DC Offset
Order 2 · even-order
|f_A − f_B| → 0 Hz (direct-conv)
Danger: In-band in zero-IF receivers. Saturates baseband.
Suppression: High IIP2, DBM, factory calibration, AC coupling.
Suppression: High IIP2, DBM, factory calibration, AC coupling.
LO Harmonic Spurs
Order 1+n · (m=1,n=2,3...)
f_RF − n·f_LO → IF
Concern: LO harmonics mix with any RF signal.
Suppression: LO filter, balanced LO (suppresses even harmonics), DBM.
Suppression: LO filter, balanced LO (suppresses even harmonics), DBM.
RF Harmonic Spurs
Order m+1 · (m=2,3...,n=1)
m·f_RF − f_LO → IF
Concern: Strong blockers generate RF harmonics inside mixer.
Suppression: RF bandpass filter, high IIP3, DBM (cancels even m).
Suppression: RF bandpass filter, high IIP3, DBM (cancels even m).
IF Feedthrough
Order 1 · (m=0,n=1 or m=1,n=0)
f_LO or f_RF appears at IF
Concern: Strong LO/RF can saturate IF amplifier.
Suppression: DBM (LO-to-IF isolation), IF bandpass filter.
Suppression: DBM (LO-to-IF isolation), IF bandpass filter.
LO Self-Mix (DC)
Order 2 · LO leakage path
f_LO − f_LO = 0
Concern: Time-varying DC offset in zero-IF architecture.
Suppression: AC coupling, adaptive DC cancellation, improve LO-RF isolation.
Suppression: AC coupling, adaptive DC cancellation, improve LO-RF isolation.
// Interactive Tool
Spur Table Generator
Enter your RF, LO and IF frequencies. The calculator generates every mixing product up to the selected maximum order, highlights any that fall within your IF passband, and colour-codes by severity.
// Spur Table Calculator — m×f_RF ± n×f_LO
MHz
MHz
MHz
MHz
| m,n | Order | Formula | f_spur (MHz) | Status | Note |
|---|
// Visualisation
Visualising Spurs on the Spectrum
The spectrum canvas below shows the RF input, LO, and the key mixing products at the IF output. The wanted IF is shown in green; spurs landing in the IF passband are shown in red; spurs outside the passband in yellow. The passband window is drawn around f_IF.
// Hardware Effect
Mixer Topology vs Spur Cancellation
The choice of mixer topology directly determines which spur families are suppressed by the balanced structure. Understanding this cancellation is essential for frequency planning.
| Spur Type | Unbalanced | Single-Balanced | Double-Balanced | Triple-Balanced |
|---|---|---|---|---|
| Image (1,1) | Present | Present | Present | Present |
| IM3 (2,1)/(1,2) | Present | Reduced | Some suppression | Better suppression |
| LO (0,1) at IF | Strong | Suppressed (20–30 dB) | Suppressed (30–50 dB) | Suppressed (40–60 dB) |
| LO 2nd harmonic (0,2) | Present | Depends on which port balanced | Cancelled | Cancelled |
| RF 2nd harmonic (2,0) | Present | Depends on which port balanced | Cancelled | Cancelled |
| Half-IF (2,2) | Strong | Partial | Significantly suppressed | Well suppressed |
| IM2 (even-order) | Strong | Partial | High suppression (IIP2) | Highest suppression |
| (3,1)/(1,3) spurs | Present | Present | Present (odd×odd survive) | Reduced |
| IF at IF (DC in ZIF) | Strong | Reduced | Reduced | Best isolation |
The Golden Rule of DBM spur cancellation: A double-balanced mixer (DBM) cancels products where m+n is even (even-order products cancel) when the cancellation is driven by balanced RF and LO ports. Only products where m is odd AND n is odd survive. The first non-cancelled product after (1,1) is (1,3) or (3,1) — order 4. This dramatically improves the spur environment vs single-balanced or unbalanced mixers.
// Engineering Solutions
Spur Suppression Techniques
1. Frequency Planning
The most powerful and cheapest spur suppression technique is choosing frequencies so that dangerous spurs fall outside the IF passband. Generate the complete spur table for the proposed frequency plan before committing to hardware. Key rules:
Frequency Planning Guidelines
Rule 1 — Avoid harmonic relationships: f_RF and f_LO should not be related by simple integer ratios. If f_RF/f_LO = m/n with small m,n, a spur of order m+n falls at DC.Rule 2 — Maximise half-IF clearance: Ensure no significant interferer sits at f_LO ± f_IF/2. Choose f_IF to place the half-IF outside congested spectrum bands.
Rule 3 — High-side vs low-side injection: Check both options — one may have a cleaner spur environment for your band plan.
Rule 4 — First IF as high as possible: Higher f_IF moves the image further from the signal (easier to filter) and moves the half-IF source further outside the preselector (easier to reject).
2. Preselector Filtering
A bandpass filter before the mixer (preselector) is the primary defence against image frequency and RF harmonic spurs. It attenuates any signal that could mix with LO harmonics to produce a spurious IF. Requirements:
— Adequate rejection at f_image (typically 50–70 dB for cellular, 60–80 dB for high-performance receivers)
— Low insertion loss in the desired receive band (loss directly degrades system noise figure)
— For wideband frequency-agile receivers: a tunable filter or switchable filter bank
— Adequate rejection at f_image (typically 50–70 dB for cellular, 60–80 dB for high-performance receivers)
— Low insertion loss in the desired receive band (loss directly degrades system noise figure)
— For wideband frequency-agile receivers: a tunable filter or switchable filter bank
3. LO Filtering and Harmonic Suppression
A lowpass or bandpass filter on the LO path attenuates LO harmonics before they reach the mixer. A sinusoidal LO (filtered from a VCO output) has far lower harmonic content than a square-wave LO. However, sinusoidal LO drive results in higher mixer conversion loss in passive mixers (which prefer rail-to-rail square waves for sharp switching). The tradeoff must be evaluated for each design.
4. Balanced Mixer Topology (DBM / Triple-Balanced)
Choosing a double-balanced or triple-balanced mixer topology is the single highest-leverage hardware decision for spur reduction. DBM eliminates all even-order products and dramatically improves IIP2, suppressing the half-IF spur and IM2 simultaneously. For direct-conversion receivers, a double-balanced Gilbert cell is mandatory — no other topology provides sufficient IIP2.
5. High IIP3 and IIP2
Every 1 dB improvement in IIP3 reduces IM3 spurs by 2 dB. Every 1 dB improvement in IIP2 reduces IM2 and half-IF spurs by 1 dB. The most reliable ways to improve these:
— Higher LO drive level (for passive diode mixers — IIP3 ≈ LO_power − 10 dB)
— More quiescent current in the transconductance stage (for active Gilbert cell mixers)
— Degeneration resistance at the RF transistor source/emitter (linearises Gm at cost of gain and NF)
— Higher LO drive level (for passive diode mixers — IIP3 ≈ LO_power − 10 dB)
— More quiescent current in the transconductance stage (for active Gilbert cell mixers)
— Degeneration resistance at the RF transistor source/emitter (linearises Gm at cost of gain and NF)
6. IF Filtering
A narrowband IF filter after the mixer attenuates all spurs that fall outside the IF passband. A SAW or BAW IF filter (in superheterodyne receivers) typically provides 40–60 dB of out-of-band rejection. Spurs that land within the IF filter passband cannot be removed — these are the truly dangerous spurs requiring the techniques above.
7. Digital Spur Cancellation
In modern software-defined receivers, known spurs (especially the LO self-mix DC offset and predictable harmonic spurs) can be estimated and cancelled in the digital domain. A reference sample of the LO signal is processed through a model of the spur generation mechanism, and the result is subtracted from the digitised baseband signal. This is most effective for the DC offset (LO self-mix) and for spurs where the spur frequency and phase are precisely known.
// Practical Summary
Practical Design Rules
These rules cover the practical decisions an RF engineer makes when designing a new receiver or transmitter:
Receiver Design Checklist — Spur Avoidance
① Generate the full spur table for your proposed f_RF, f_LO, f_IF before layout.Use the interactive tool above for orders 1–7. Focus on products that fall within f_IF ± BW/2.
② Check the image frequency against the preselector rejection.
Image rejection = preselector attenuation at f_image. Target ≥ 60 dB for most applications.
③ Check the half-IF spur source at f_LO + f_IF/2.
Is there any significant signal at this frequency? If yes, either change IF or use high-IIP2 mixer.
④ Check (3,1) and (1,3) products — the lowest-order survivors in a DBM.
(3,1): |3·f_RF − f_LO| = f_IF? → f_RF = (f_LO + f_IF)/3
(1,3): |f_RF − 3·f_LO| = f_IF? → f_RF = 3·f_LO ± f_IF
⑤ Verify LO harmonic rejection — do any harmonics of f_LO, when mixed with f_RF,
produce output within the IF band? n·f_LO ± f_RF = f_IF for n=2,3,4?
⑥ Budget the cascaded IIP3 with the Friis linearity cascade formula.
The mixer should not dominate cascaded IIP3 for the desired dynamic range.
⑦ For direct-conversion: Verify IIP2 > +50 dBm after calibration,
and ensure LO-to-RF isolation > 40 dB to control DC offset level.
Complete Spur Analysis — 900 MHz Receiver Example
Design: f_RF = 900 MHz, f_LO = 800 MHz, f_IF = 100 MHz, IF BW = ±5 MHz
Mixer: Double-balanced passive ring, IIP3 = +10 dBm, IIP2 = +40 dBm
Image: f_image = 2×800 − 900 = 700 MHz → need 60 dB preselector rejection at 700 MHz ✓ (200 MHz from wanted, achievable)
Half-IF spur source: f_LO + f_IF/2 = 800 + 50 = 850 MHz → 50 MHz from wanted signal → preselector must attenuate 850 MHz by >40 dB → achievable with a good BPF
(3,1) check: |3×900 − 800| = |2700−800| = 1900 MHz → not at IF ✓
(1,3) check: |900 − 3×800| = |900−2400| = 1500 MHz → not at IF ✓
(3,2) check: |3×900 − 2×800| = |2700−1600| = 1100 MHz → not at IF ✓
(2,3) check: |2×900 − 3×800| = |1800−2400| = 600 MHz → not at IF ✓
Result: This frequency plan has no low-order spurs within the IF band. The preselector must provide 60 dB at 700 MHz (image) and 40 dB at 850 MHz (half-IF source). A 4-pole LC bandpass filter centred at 900 MHz with 30 MHz bandwidth achieves both.
Mixer: Double-balanced passive ring, IIP3 = +10 dBm, IIP2 = +40 dBm
Image: f_image = 2×800 − 900 = 700 MHz → need 60 dB preselector rejection at 700 MHz ✓ (200 MHz from wanted, achievable)
Half-IF spur source: f_LO + f_IF/2 = 800 + 50 = 850 MHz → 50 MHz from wanted signal → preselector must attenuate 850 MHz by >40 dB → achievable with a good BPF
(3,1) check: |3×900 − 800| = |2700−800| = 1900 MHz → not at IF ✓
(1,3) check: |900 − 3×800| = |900−2400| = 1500 MHz → not at IF ✓
(3,2) check: |3×900 − 2×800| = |2700−1600| = 1100 MHz → not at IF ✓
(2,3) check: |2×900 − 3×800| = |1800−2400| = 600 MHz → not at IF ✓
Result: This frequency plan has no low-order spurs within the IF band. The preselector must provide 60 dB at 700 MHz (image) and 40 dB at 850 MHz (half-IF source). A 4-pole LC bandpass filter centred at 900 MHz with 30 MHz bandwidth achieves both.