Analog Receivers¶

Exam-ready answer

A superheterodyne receiver converts every selected RF station to one fixed intermediate frequency (IF), where most gain and selectivity are obtained. The fixed frequency permits stable high gain and sharp, repeatable filters over the complete tuning range.

The antenna/input network captures many RF signals. The tuned RF amplifier or preselector selects the wanted band, improves sensitivity with low added noise and rejects the image. The local oscillator (LO) tracks tuning. The nonlinear mixer produces sum and difference frequencies, and the IF filter selects

The fixed IF filter/amplifier supplies most voltage gain and adjacent-channel selectivity. For conventional AM broadcast, \(455\,\text{kHz}\) is common, though not universal. The envelope detector recovers audio and a slowly varying AGC component. The AF amplifier drives the loudspeaker. Automatic gain control (AGC) reduces RF/IF gain for strong stations and increases it for weak ones, keeping output usable and avoiding overload; its time constant is slow relative to audio so it follows average carrier strength rather than modulation.

For high-side injection, \(f_{LO}=f_s+f_{IF}\). An undesired input on the other side of the LO also gives the same IF:

For low-side injection, \(f_{LO}=f_s-f_{IF}\) and \(f_{image}=f_s-2f_{IF}\). Once desired and image signals reach the mixer, both become the same IF, so the IF filter cannot separate them; RF preselection must reject the image beforehand. A higher first IF separates the image farther and improves rejection, whereas a lower IF makes narrow channel filtering easier. Dual conversion uses a high first IF followed by a lower second IF to obtain both advantages.

For \(f_s=1.000\,\text{MHz}\), \(f_{IF}=455\,\text{kHz}\) and high-side injection, \(f_{LO}=1.455\,\text{MHz}\) and \(f_{image}=1.910\,\text{MHz}\). Both satisfy \(|f-f_{LO}|=455\,\text{kHz}\). The receiver's limitations are mixer spurs, LO leakage/drift, image response and greater complexity than TRF; its advantages are uniform bandwidth, high sensitivity, strong selectivity and easy multistage gain. The complete signal path preserves the AM carrier plus both sidebands through an IF bandwidth of at least about \(2f_{m(max)}\) before detection.

Practice target: 16-18 minutes; draw and label every block, derive both IF and image relations, and explain the high-IF/low-IF trade-off.

Exam-ready answer

A tuned radio-frequency (TRF) receiver selects and amplifies the desired station directly at its incoming RF; it has no mixer, local oscillator or intermediate-frequency stage.

The antenna receives many stations. Two or more RF amplifier stages contain parallel resonant circuits tuned to the wanted carrier. Their capacitors are mechanically/electronically ganged so all resonances track together. After RF gain and selectivity, a diode envelope detector recovers the AM message, and an AF voltage/power amplifier drives the loudspeaker. For each tuned circuit,

where \(L\) is henry, \(C\) is farad, \(Q\) is loaded quality factor and \(B\) is \(3\,\text{dB}\) bandwidth in hertz.

Advantages are a simple signal path, low component count, no internally generated image frequency and useful performance over a narrow or fixed-frequency range. Limitations are difficult simultaneous tracking of several RF resonators, changing absolute bandwidth when \(Q\) is nearly constant, limited stable gain at high RF because of stray feedback, and nonuniform selectivity across a wide tuning band. For example, with \(Q=100\), bandwidth is \(5.5\,\text{kHz}\) at \(550\,\text{kHz}\) but \(16\,\text{kHz}\) at \(1.6\,\text{MHz}\), so one end may cut wanted sidebands while the other admits adjacent stations. A superheterodyne avoids this by converting every station to one fixed IF, but adds mixer/image complexity. TRF remains useful for simple fixed-frequency or narrow-range receivers.

Practice target: 7-8 minutes; draw the cascaded tuned stages and use \(B=f_0/Q\) to explain the central limitation.

Exam-ready answer

In a superheterodyne receiver the mixer and IF filter use

Given desired signal \(f_s=1.000\,\text{MHz}\), IF \(f_{IF}=455\,\text{kHz}=0.455\,\text{MHz}\) and high-side injection, the LO lies above the wanted station:

The image lies the same IF distance on the opposite side of this LO. Therefore

or directly \(f_{image}=f_s+2f_{IF}=1.000+2(0.455)=1.910\,\text{MHz}\).

The essential numerical check is

Hence both the wanted station and image become \(455\,\text{kHz}\) after mixing. The IF filter cannot distinguish them; a tuned RF preselector before the mixer must attenuate \(1.910\,\text{MHz}\). Image separation from the desired station is \(2f_{IF}=910\,\text{kHz}\). A higher IF increases this separation and eases RF image rejection, while a lower IF permits a narrower fixed IF filter and better adjacent-channel selectivity. If low-side injection had instead been specified, the formulas would be \(f_{LO}=f_s-f_{IF}\) and \(f_{image}=f_s-2f_{IF}\), provided the resulting frequencies are positive and within the receiver plan.

Practice target: 7-8 minutes; write the injection relation first, retain common units, and verify both inputs produce the same IF.

Exam-ready answer

A superheterodyne FM receiver selects an FM station, converts it to a fixed IF and recovers the message from instantaneous-frequency deviation rather than amplitude.

The RF amplifier/preselector gives low-noise gain and rejects image/adjacent signals. The mixer and LO satisfy \(f_{IF}=|f_{LO}-f_s|\); \(10.7\,\text{MHz}\) is a common FM-broadcast IF. The IF filter/amplifier supplies most gain and passes the complete FM channel, approximately \(2(\Delta f+f_{m(max)})\) by Carson's rule. A filter too narrow removes significant sidebands and distorts audio.

The limiter clips amplitude variations caused by noise and fading while retaining zero crossings/frequency deviation, because ideal FM has constant envelope. It is particularly necessary before amplitude-sensitive slope and Foster-Seeley discriminators; ratio and PLL detectors have more inherent AM rejection. The discriminator has an S-curve: output is zero at center, positive on one side, negative on the other and approximately proportional to \(f_i-f_c\) in its linear region. Implementations include Foster-Seeley, ratio, quadrature and PLL detectors.

At the transmitter, pre-emphasis boosts high audio frequencies because post-detection FM noise rises strongly with frequency. The receiver de-emphasis network is a complementary first-order RC low-pass with

commonly \(50\,\mu\text{s}\) or \(75\,\mu\text{s}\) depending on the standard. It restores the original audio response and attenuates high-frequency noise. For \(R=75\,\text{k}\Omega\) and \(C=1\,\text{nF}\), \(RC=75\,\mu\text{s}\).

Automatic frequency control (AFC) uses the discriminator's slowly varying DC error to correct LO drift through a varactor or controlled oscillator. Its polarity pulls the IF back to center. AFC controls frequency, whereas AGC controls gain; they are not interchangeable. The recovered audio then passes through coupling/filtering and the AF amplifier to the speaker.

For broadcast \(\Delta f=75\,\text{kHz}\) and \(f_{m(max)}=15\,\text{kHz}\), Carson bandwidth is \(180\,\text{kHz}\), motivating an IF/channel near \(200\,\text{kHz}\). FM offers limiting, capture effect and high fidelity above threshold, but uses more bandwidth and circuitry than AM. Below the FM threshold, output SNR collapses rapidly; discriminator misalignment, oscillator drift and multipath also cause distortion. Proper RF preselection, IF bandwidth, limiting, de-emphasis and AFC are therefore all required for reliable reception.

Practice target: 16-18 minutes; draw the complete chain and explain limiter, discriminator, de-emphasis and AFC as four distinct functions.

Exam-ready answer

Sensitivity is a receiver's ability to produce a specified usable output from a weak RF input. It is stated as minimum input power/voltage for a required output SNR, SINAD or BER. It depends mainly on antenna/interface loss, first-stage noise figure, gain and detector threshold; lower required input power means better sensitivity.

Selectivity is the ability to accept the wanted channel while rejecting adjacent or other unwanted frequencies. It is determined mainly by RF/IF filter bandwidth, skirt slope or shape factor, image rejection and dynamic range. Narrower, steeper filters generally improve adjacent-channel selectivity.

Fidelity is the ability to reproduce the original message without significant amplitude, phase or nonlinear distortion. It requires the receiver passband to include all wanted modulation components with approximately flat magnitude and sufficiently linear phase. For AM audio bandwidth \(f_m\), the IF must pass carrier and both sidebands, so \(B_{IF}\gtrsim2f_m\); FM must pass its significant Carson sidebands.

The main trade-off is that narrowing IF bandwidth reduces admitted noise \(N=kTB\) and adjacent-channel interference, improving sensitivity/selectivity, but a bandwidth narrower than the wanted spectrum cuts sidebands and reduces fidelity. Widening it improves waveform reproduction but admits more noise and neighboring channels. Likewise, very high front-end gain can improve weak-signal response yet cause compression/intermodulation in the presence of strong signals. For a \(455\,\text{kHz}\) AM IF carrying \(5\,\text{kHz}\) audio, about \(10\,\text{kHz}\) passband is required; an equivalent single-tuned \(Q\) is \(f_0/B\approx45.5\). A \(5\,\text{kHz}\) filter would sound muffled, while a much wider filter would admit unnecessary noise. Thus receiver design balances minimum usable signal, rejection and required message bandwidth rather than maximizing any one term alone.

Practice target: 7-8 minutes; define all three terms with a measurable criterion and explain the IF-bandwidth trade-off using one numerical example.