7.3 Superheterodyne Receiver Design, IF Selection, and Image Rejection

7.3 Superheterodyne Receiver Design, IF Selection, and Image Rejection

In the early era of wireless communications, receivers relied on Tuned Radio Frequency (TRF) architectures, where multiple cascaded amplifier stages were tuned synchronously to the incoming radio frequency. Ganged capacitors could tune several stages together, but the circuits needed careful alignment to track across a band. Selectivity also varied with frequency: at approximately constant $Q$, the absolute bandwidth $\Delta f=f_0/Q$ increases as the tuned frequency rises.

In 1918, Edwin Armstrong solved these limitations by inventing the superheterodyne (supersonic heterodyne) receiver. By translating received RF to a fixed Intermediate Frequency (IF), above or below the received frequency, gain and selectivity can be optimized at that fixed IF. Today, the superheterodyne remains the gold standard for high-performance amateur radio communications.

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1. Superheterodyne Architecture and Functional Blocks

The operation of a classic single-conversion superheterodyne receiver can be traced through its sequential functional stages:

Antenna --> RF filter/amplifier --> mixer (+ LO) --> selected IF filter/gain
                                                               |
                           +-----------------------------------+
                           | CW A1A / SSB J3E: product detector (+ BFO)
                           | AM A3E: envelope or synchronous detector
                           | FM F3E: limiter --> discriminator/PLL detector
                           v
                  audio filter --> squelch gate --> audio amplifier --> speaker
AGC controls appropriate earlier gain stages; squelch controls audio muting.

1. RF Preselector: A passive bandpass filter or sub-octave preselector situated immediately at the antenna terminal. Its sole duty is to restrict the bandwidth of RF energy reaching the active stages, attenuating powerful out-of-band broadcast transmitters and, crucially, rejecting the receiver's image frequency.
2. Low-Noise RF Amplifier (LNA): A high-linearity, small-signal amplifier (typically using dual-gate MOSFETs, low-noise JFETs, or PHEMTs) that establishes the receiver's overall Noise Figure ($NF$). In high-performance HF receivers where atmospheric noise dominates, this stage is often switchable (e.g. $+10\text{ dB}$ preamp, bypass, or $-6/-12/-18\text{ dB}$ step attenuator) to prevent front-end overload in the presence of strong signals.
3. First Mixer and Local Oscillator (LO): Combines the incoming RF signal with the agile, synthesised LO carrier, translating the signal to the fixed intermediate frequency ($f_{\text{IF}} = |f_{\text{RF}} - f_{\text{LO}}|$).
4. Post-Mixer Roofing Filter: A narrow quartz crystal or $LC$ filter placed immediately at the mixer output. It restricts the bandwidth of signals entering the downstream IF amplifiers to between $250\text{ Hz}$ (for CW) and $3\text{ kHz}$ to $15\text{ kHz}$ (for SSB/AM), protecting subsequent stages from off-frequency overload.
5. Intermediate Frequency (IF) Amplifier Strip: The "engine room" of the receiver. It provides the vast majority of the receiver's overall system gain ($80\text{ dB}$ to $100\text{ dB}$). Fixed-frequency operation lets each stage be optimized for one passband. Cascaded high gain still requires careful layout, shielding and supply decoupling to prevent unwanted feedback and self-oscillation. This stage is governed by the Automatic Gain Control (AGC) loop.
6. Demodulator / Product Detector: Translates the modulated IF signal down to baseband audio:
   • For SSB and CW, a product detector (a balanced mixer) multiplies the IF signal with a local carrier generated by a Beat Frequency Oscillator (BFO).
   • For AM, an envelope diode detector or synchronous detector is used.
   • For FM, a quadrature detector, phase-locked loop (PLL), or discriminator demodulates frequency shifts.
7. Audio Amplifier and Active Filter: Boosts the baseband audio signal to drive headphones or an external loudspeaker, incorporating low-pass and notch filtering to eliminate audio hiss and interfering heterodyne whistles.

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Squelch and Automatic Gain Control

Squelch mutes the audio when no acceptable received signal is present. An FM noise-squelch circuit can monitor high-frequency detector noise; carrier-operated squelch uses a signal-strength condition. The threshold determines when audio opens. Set it just high enough to silence idle-channel noise; too high a setting hides weak stations. Opening/monitoring the squelch helps when searching for weak signals.

AGC instead adjusts RF/IF gain over changing signal strengths. It helps maintain a useful audio level and avoids overload in the controlled stages. Squelch is not a sensitivity improvement or a substitute for AGC.

2. Image Frequency Mechanics and Image Rejection Ratio (IRR)

A real mixer produces sum and difference components. In the usual difference-product case with $f_{\text{LO}}>f_{\text{IF}}$, RF signals one IF above or below the LO both produce the selected IF. These are:

1. The Desired Signal Frequency ($f_{\text{sig}}$).
2. The Unwanted Image Frequency ($f_{\text{image}}$).

Mathematical Derivation of Image Frequency

Depending on whether the local oscillator is tuned above or below the incoming signal frequency, the mixer operates in one of two modes:

1. High-Side Local Oscillator Injection ($f_{\text{LO}} > f_{\text{sig}}$)

The LO is tuned above the desired signal: $f_{\text{LO}} = f_{\text{sig}} + f_{\text{IF}}$. An unwanted signal entering the mixer at an even higher frequency can also mix with the LO to yield the difference $f_{\text{IF}}$:

$$|f_{\text{image}} - f_{\text{LO}}| = f_{\text{IF}} \implies f_{\text{image}} = f_{\text{LO}} + f_{\text{IF}}$$

Substituting $f_{\text{LO}} = f_{\text{sig}} + f_{\text{IF}}$ into the image equation:

$$f_{\text{image}} = (f_{\text{sig}} + f_{\text{IF}}) + f_{\text{IF}} = f_{\text{sig}} + 2 f_{\text{IF}}$$

2. Low-Side Local Oscillator Injection ($f_{\text{LO}} < f_{\text{sig}}$)

The LO is tuned below the desired signal: $f_{\text{LO}} = f_{\text{sig}} - f_{\text{IF}}$. If $f_{\text{LO}}>f_{\text{IF}}$, an unwanted positive-frequency signal below the LO also gives the difference $f_{\text{IF}}$:

$$|f_{\text{LO}} - f_{\text{image}}| = f_{\text{IF}} \implies f_{\text{image}} = f_{\text{LO}} - f_{\text{IF}}$$

Substituting $f_{\text{LO}} = f_{\text{sig}} - f_{\text{IF}}$ into the image equation:

$$f_{\text{image}} = (f_{\text{sig}} - f_{\text{IF}}) - f_{\text{IF}} = f_{\text{sig}} - 2 f_{\text{IF}}$$

Image Rejection and Receiver Architecture

In a single real mixer path, the desired signal and its image overlap at the same IF. A later frequency-selective filter, including DSP on that one path, cannot separate them. RF filtering before the mixer must therefore attenuate the image.

Image Rejection Ratio (IRR)

The Image Rejection Ratio (IRR) quantifies the receiver's ability to reject signals at the image frequency. It is defined as the ratio of receiver sensitivity at the image frequency to sensitivity at the desired frequency, expressed in decibels:

$$\text{IRR (dB)} = 10 \log_{10}\left( \frac{P_{\text{in(image)}}}{P_{\text{in(desired)}}} \right) \quad \text{for equal IF output power}$$

For a single tuned $LC$ preselector circuit with loaded quality factor $Q_L$, the image rejection ratio can be calculated analytically by:

$$\text{IRR} = \sqrt{1 + Q_L^2 \left( \frac{f_{\text{image}}}{f_{\text{sig}}} - \frac{f_{\text{sig}}}{f_{\text{image}}} \right)^2}$$

In decibels:

$$\text{IRR (dB)} = 20 \log_{10}\left[ \sqrt{1 + Q_L^2 \left( \frac{f_{\text{image}}}{f_{\text{sig}}} - \frac{f_{\text{sig}}}{f_{\text{image}}} \right)^2} \right]$$

The required IRR depends on the strength of unwanted signals and the tolerated interference. Compare the receiver specification and measured rejection at the relevant tuning frequency; the IF value alone does not set the rejection.

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3. The Choice of Intermediate Frequency: The "IF Dilemma"

Selecting the intermediate frequency represents one of the most famous compromises in radio frequency engineering, universally known as the IF dilemma.

                    THE IF DILEMMA TRADE-OFF
                    
   High Intermediate Frequency       Low Intermediate Frequency
   (e.g., 70 MHz Up-Conversion)      (e.g., 455 kHz Down-Conversion)
   
   +--------------------------+      +--------------------------+
   | EXCELLENT Image Rejection|      | POOR Image Rejection     |
   | Image is 140 MHz away;   |      | Image is 910 kHz away;   |
   | RF filtering is easier   |      | RF filtering is harder   |
   | at this separation.      |      | at this separation.      |
   +--------------------------+      +--------------------------+
                 |                                 |
   +--------------------------+      +--------------------------+
   | POOR Adjacent Selectivity|      | EXCELLENT Selectivity    |
   | High Q needed for narrow |      | Low Q needed for narrow  |
   | bandwidth; crystal filter|      | bandwidth; steep-skirted |
   | skirts are broad.        |      | crystal/mechanical filter|
   +--------------------------+      +--------------------------+

The Trade-Off Analysis

1. If a HIGH Intermediate Frequency is chosen (e.g. $70.0\text{ MHz}$):

   • Image Separation: $2 f_{\text{IF}} = 2 \times 70\text{ MHz} = 140\text{ MHz}$. For any HF signal between $1.8\text{ MHz}$ and $30\text{ MHz}$, the image lies between $141.8\text{ MHz}$ and $170\text{ MHz}$. This wide separation permits a fixed antenna low-pass filter to attenuate the image while passing HF signals; the rejection depends on its design and construction.
   • Selectivity Defect: Building a narrow $2.4\text{ kHz}$ SSB filter at $70\text{ MHz}$ requires an extreme fractional filter $Q$ ($Q = 70\text{ MHz} / 2.4\text{ kHz} \approx 29,167$). Quartz crystal filters at this frequency have broad, sloppy skirts (high shape factors) and significant insertion loss.

2. If a LOW Intermediate Frequency is chosen (e.g. $455\text{ kHz}$):

   • Selectivity Advantage: At $455\text{ kHz}$, a $2.4\text{ kHz}$ filter requires a fractional $Q$ of only $455 / 2.4 \approx 190$. Multi-pole mechanical or ceramic filters easily achieve ultra-steep, "brick-wall" filter skirts with shape factors approaching $1.5$.
   • Image Defect: Image separation is only $2 \times 455\text{ kHz} = 910\text{ kHz}$. On the $28\text{ MHz}$ (10-meter) band, an image only $910\text{ kHz}$ away represents a frequency deviation of just $3.25\%$. Strong image rejection therefore requires more selective RF filtering, often using several tuned circuits. Tracking can be manual or electronic; a motor is not an electrical requirement.

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4. The Dual-Conversion Superheterodyne Solution

A dual-conversion superheterodyne can combine a high first IF for easier first-image filtering with a low second IF for narrow channel filtering:

+-----------+    RF    +-----------+  High 1st IF   +----------+  Low 2nd IF   +----------+  Audio
| Preselect |--->|X|-->| 1st IF    |------>|X|--------->| 2nd IF   |------>|X|--------->...
+-----------+     ^    | Roofing   |        ^           | Narrow   |        ^
                  |    | Filter    |        |           | Filter   |        |
                +---+  +-----------+      +---+         +----------+      +---+
                |LO1|                     |LO2|                           |BFO|
                +---+                     +---+                           +---+

1. First Conversion (Up-Conversion to High 1st IF): The incoming RF signal is mixed with a synthesised LO to translate it to a high first IF (e.g. $70.455\text{ MHz}$). With high-side injection, the first image is $140.91\text{ MHz}$ above the wanted RF. A suitable fixed antenna low-pass filter can strongly attenuate it.
2. First Roofing Filter: The signal passes through a crystal roofing filter at $70.455\text{ MHz}$ with a bandwidth of $6\text{ kHz}$ to $15\text{ kHz}$, protecting downstream stages.
3. Second Conversion (Down-Conversion to Low 2nd IF): A second mixer driven by a fixed crystal oscillator ($f_{\text{LO2}} = 70.000\text{ MHz}$) translates the signal down to a low second IF: $70.455 - 70.000 = \mathbf{455\text{ kHz}}$.
4. Second IF Selectivity: At $455\text{ kHz}$, steep mechanical or multi-pole crystal ladder filters provide razor-sharp adjacent-channel selectivity with shape factors under $1.6$.

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5. Roofing Filter Function and Placement

The roofing filter is a narrow crystal ladder filter positioned immediately after the first mixer and IF diplexer network.

• The Overload Mechanism: Consider a wanted CW signal of $0.1\ \mu\text{V}$ and an interferer of $100\text{ mV}$, measured RMS across a $50\ \Omega$ receiver input. These are approximately $-127\text{ dBm}$ and $-7\text{ dBm}$, a $120\text{ dB}$ difference. Let the interferer be $3\text{ kHz}$ away.
• Without a Narrow Roofing Filter: A $15\text{ kHz}$ first-IF passband can admit both signals to the following IF amplifier. Depending on gain and headroom, the interferer can cause compression, cross-modulation or AGC gain reduction that masks the wanted signal.
• With a Narrow Roofing Filter: Use its specified response at the interferer's offset. If it supplies $60\text{ dB}$ more attenuation there than at the wanted frequency, the $120\text{ dB}$ difference falls to $60\text{ dB}$ before later IF/ADC stages. This improves their headroom, but the RF amplifier and first mixer ahead of the filter still need adequate strong-signal performance.

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6. Architecture Comparison Table

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7. Worked Engineering Examples

Worked Example 7.3.1: Image Frequency Calculations for High-Side and Low-Side Injection

An amateur communications receiver utilizes a first intermediate frequency of $f_{\text{IF}} = 9.000\text{ MHz}$. Calculate the local oscillator frequency and the resulting image frequency under two different tuning scenarios:

1. Scenario A: The receiver is tuned to $f_{\text{sig}} = 7.150\text{ MHz}$ on the 40-meter band using high-side local oscillator injection:

$$f_{\text{LO}} = f_{\text{sig}} + f_{\text{IF}} = 7.150\text{ MHz} + 9.000\text{ MHz} = 16.150\text{ MHz}$$ $$f_{\text{image}} = f_{\text{sig}} + 2 f_{\text{IF}} = 7.150\text{ MHz} + (2 \times 9.000\text{ MHz}) = 25.150\text{ MHz}$$

The image lies at $25.150\text{ MHz}$. An unwanted signal there could produce the same IF, so the 40-metre preselector must attenuate that frequency.

1. Scenario B: The receiver is tuned to $f_{\text{sig}}=14.250\text{ MHz}$ using low-side injection:

$$f_{\text{LO}}=14.250-9.000=5.250\text{ MHz}$$

The expression $14.250-2(9.000)=-3.750\text{ MHz}$ identifies a positive image at $3.750\text{ MHz}$. Check its actual mixing product:

$$f_{\text{image}}+f_{\text{LO}}=3.750+5.250=9.000\text{ MHz}$$

The wanted signal uses the difference $14.250-5.250=9.000\text{ MHz}$; the image uses the sum. Their separation is $14.250-3.750=10.500\text{ MHz}$, not $2f_{\text{IF}}$. RF preselection must reject the $3.750\text{ MHz}$ response.

Worked Example 7.3.2: Calculating Image Rejection Ratio (IRR) from Preselector Quality Factor

A receiver front-end consists of a single parallel $LC$ resonant preselector tuned to $f_{\text{sig}} = 14.100\text{ MHz}$. The loaded quality factor of the tank circuit is $Q_L = 40$. The receiver uses a low IF of $f_{\text{IF}} = 455\text{ kHz}$ with high-side LO injection.

1. Calculate the image frequency:

$$f_{\text{image}} = f_{\text{sig}} + 2 f_{\text{IF}} = 14.100\text{ MHz} + (2 \times 0.455\text{ MHz}) = 15.010\text{ MHz}$$

1. Calculate the normalized frequency detuning parameter ($y$):

$$y = \frac{f_{\text{image}}}{f_{\text{sig}}} - \frac{f_{\text{sig}}}{f_{\text{image}}} = \frac{15.010}{14.100} - \frac{14.100}{15.010} = 1.0645 - 0.9394 = 0.1251$$

1. Calculate the Image Rejection Ratio:

$$\text{IRR} = \sqrt{1 + (Q_L \cdot y)^2} = \sqrt{1 + (40 \times 0.1251)^2} = \sqrt{1 + (5.004)^2} = \sqrt{1 + 25.04} = \sqrt{26.04} \approx 5.103$$

1. Express the IRR in decibels:

$$\text{IRR (dB)} = 20 \log_{10}(5.103) \approx 14.16\text{ dB}$$

Analysis: The single tank attenuates this image by only $14.16\text{ dB}$. An image signal more than $14.16\text{ dB}$ stronger than the wanted signal would still give a larger IF output. More selective RF filtering or a different frequency plan can improve rejection. The result demonstrates this particular preselector's limitation; it does not prove that all single-conversion HF receivers must use one architecture or tuning mechanism.