Reciprocal Mixing - how LO phase noise affects selectivity

Reciprocal mixing occurs in a super-heterodyne receiver when the noise sidebands of the Local Oscillator mix with strong signals,   that are close in frequency to the wanted signal,  producing unwanted noise products at the intermediate frequency and degrading the receiver sensitivity.

A typical receiver front end down-conversion stage is shown in Figure 2.1

Figure 2.1 - Receiver 1st Mixer

Here the incoming RF signal is mixed with the local oscillator signal to produce the output at the IF.   Figure 2.2 shows the ideal relationships between the signal frequencies.

Figure 2.2  Down Conversion Frequency Relationships

For this discussion we assume that the LO frequency is above the RF frequency,  but the same effects occur if low-side tuning is used.     To tune the receiver to f_IF  the local oscillator is set to a frequency 

f_LO  = f_RF  + f_IF

The image filter in front of the mixer has the job of attenuating signals at the image frequency ( f_image  = f_LO  + f_IF  = f_RF  + 2 f_IF ) so that they will not be converted to f_IF as well.  The close-in selectivity of such a receiver is usually determined by the IF filter selectivity. 

Example: A FM radio receiver operates with 12.5kHz channel spacing.   The requirement on adjacent channel rejection (ACR) is that an unmodulated signal in the adjacent channel,  60dB stronger than the Minimum Detectable Signal (MDS),  shall not degrade the receiver sensitivity by more than 3dB.   Determine the IF Filter response needed to meet this requirement.

The attenuation of this nearby signal will be determined by the response of the IF strip (principally the IF filter).   To determine the amount of rejection required,  the response of the demodulator to this interferer spaced at 12.5kHz must be determined.   If it is found that this must be kept to X dB below the wanted signal,  then 

for 3dB degradation in sensitivity,  wanted signal = MDS + 3dB

unwanted interferer = MDS + 60dB  = wanted signal + 57dB

so the rejection required at 12.5kHz offset is 57dB plus X dB.   So to keep the unwanted signal 6dB below the wanted signal required 63dB of attenuation at 12.5kHz. 

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In the real case,  where the local oscillator has phase noise,  the spectrum of the local oscillator is not the ideal delta function as shown in Figure 2.2.   A more realistic situation is shown in Figure 2.3  

Figure 2.3 Reciprocal Mixing

Here the phase noise on the Local Oscillator is shown.   To see the effect this can have on selectivity,  imagine the a strong interfering signal close to the wanted RF signal as shown.   This unwanted signal can mix with the noise sidebands of the LO to also produce an output at the IF.   This process is called reciprocal mixing.   The noise sidebands involved are shown shaded in Figure 2.3.   

To determine the relative power produced at the IF for the wanted signal and the interferer,  firstly notice that the wanted signal is mixed with the LO carrier,  whereas the interferer is mixed with a signal with the power in the noise sidebands

P_rm = L_f(f_m) B_w  (relative to the carrier)

.. (2.1)     

B_w is the bandwidth of the IF filter.  Note this is approximate - if the phase noise curve has unusual lumps,  or if B_w is large,  or if the IF filter has an unusual shape,  then to accurately determine the noise you should integrate the curve.   See note at the end as how this can be achieved using SimPLL.

Thus at the output of the downconverter there are two signals,  the wanted signal and the unwanted reciprocally mixed noise products.   To determine the relative power levels,  note that the they are converted at relative power gains of  1 : P_rm .  Thus,  if the wanted and unwanted signals have equal powers entering the downconverter,  then the wanted signal emerges with power 1 relative to the reciprocally mixed noise which has power P_rm.    This enables the calculation of the amount of noise present in the IF strip from reciprocal mixing.

Example: A receiver local oscillator has phase noise at 25kHz offset of -100dBc.   What is the sensitivity,  relative to an in-band signal,   to an interferer spaced 25kHz from the carrier,  assuming the IF bandwidth is 15kHz.   What will be the S/N at the IF from reciprocal mixing if there is an interferer 50dB above the wanted signal and offset by 25kHz.

The sensitivity relative to the wanted signal is 

-100dBc/Hz + 10 log₁₀(15kHz) = -58.2dB

Thus for an interferer with level 50dB above the wanted signal,  the noise  power produced at the IF will be 50 - 58.2 = -8.2dB relative to the wanted signal,  so the S/N is 8.2dB.  

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As reciprocal mixing adds the unwanted signal mixed with the noise sidebands of the LO to our IF output,  any phase or frequency modulation on the interfering signal is typically lost in the process,  however amplitude modulation is preserved and modulates the added noise.   For example,  if the interferer carrier was gated on and off,  the reciprocally mixed noise would appear and disappear.

Example: For the FM radio receiver in the previous example,   determine the phase noise specification for the Local Oscillator,  assuming that the required S/N is 12dB and the IF bandwidth is 8kHz.

Under test: 

Wanted signal = MDS + 3dB
system noise = MDS - 12dB

interferer at 12.5kHz offset = MDS + 60dB
allowable noise from reciprocal mixing = MDS -12dB

We have used the fact that a 3dB degradation in sensitivity is caused by a doubling of system noise.   So,  under test,  the interferer is 57dB above the wanted signal,  and the allowable noise at the IF must be 15dB below the wanted signal,  thus the interferer must be suppressed by 72dB relative to the wanted signal.   The IF bandwidth is 10 log₁₀(8kHz) = 39dB Hz,  thus

phase noise  <=  -111dBc/Hz at 12.5kHz offset   

This shows the phase noise required to meet the ACR specification,  previously we determined the IF response needed.   If the system is phase noise limited then the IF response needs to be considerably better than determined above,  conversely if the IF response is marginal,  then clearly the phase noise specification must be significantly better than -111dBc/Hz so as not to degrade the filter response.

For designing PLL Synthesizers,  SimPLL provides the easy way to calculate P_rm  (given by equation 2.1) by numerically integrating the phase noise curve.   You can set the channel offset and bandwidth under Edit / Report Options on the main menu. 

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