**Introduction**

The need to translate frequencies up or down is a basic signal-chain need. The key component used to do this is the mixer, which essentially multiplies two signals together.

In the frequency domain, this is equivalent to producing two output signals at frequencies that are the sum and difference of the original frequencies, easily verified with this trigonometric identity (Equation 1):

In Figure 1, the desired signal at the radio frequency (RF) of the receive signal, f_{Receive} , comes to the mixer and is combined with a local oscillator (LO) frequency, f_{LO} , that adjusts so that the difference between the LO signal and f_{Receive} is a constant intermediate frequency (IF), f_{IF} . The mixer will also produce an undesired frequency that is the sum of the RF and LO frequencies, but this is filtered out.

**Figure 1**

**Single-stage downconversion**

Consider an application where the received frequency is 5,000-8,000MHz in 1MHz steps, which is mixed down to a fixed IF frequency of 500MHz. So 5,001MHz would use an LO frequency of 5,501MHz and 8,000MHz would use an LO frequency of 8,500MHz. The received signal is typically modulated with digital information and is typically translated to a lower frequency, say 500MHz.

The challenge in this application is the mixer image problem. Let’s say that the system was trying to receive a signal at 6,001MHz. To do this, the LO would be tuned to 6,501MHz to produce a 500MHz IF signal. However, an undesired signal at 7,001MHz would also produce an unwanted noise signal at 500MHz, as shown in Figure 2.

**Figure 2**

**Mixer image example**

One approach to solving this issue is to mix down in multiple stages, as in the superheterodyne architecture shown in Figure 3.

**Figure 3**

**Superheterodyne architecture**

**Image reject approach with upconversion**

One approach that enables mixing to a low IF frequency without the use of expensive filters is to use two mixers with two 90-degree phase shifts, as shown in Figure 4.

**Figure 4**

**Image reject upconversion**

First starting with signals of the form (Equations 2 and 3):

Equation 4 calculates the in-phase (I) and quadrature (Q) outputs:

Now the quadrature output has some phase shifts in it (Equation 5):

So when adding the output together, you theoretically only get the desired sideband and the undesired mixer image is completely eliminated (Equation 6):

These calculations assume that both mixers have exactly the same gain and that the phase shifts are a perfect 90 degrees. In reality, they will not be perfect and will lead to a mixer image of some level, although it will be much lower than the desired frequency.

Without loss of generality, normalize the gain and phase to the in-phase (I) output and assume that the quadrature output (Q) has some gain imbalance (θ) and phase imbalance expressed in radians (*θ* ) (Equation 7):

So if you do I+Q, you get Equation 8:

Now introduce a Taylor series expansion that is valid if * θ * is small (Equation 9):

And apply this expression in equation 9 to equation 8 to get the result of equation 10.:

The desired signal is at the frequency of f_{IF} + f_{LO} in equation 10 and can be approximated assuming *ε * is small (equation 11).

The undesired sideband is the remaining terms at frequency f_{RF} – f_{LO} and is in equation 12.

Now consider the magnitude of the sideband to the magnitude of the desired signal and convert to power to get the fundamental identity (Equation 13):

**Image reject approach with downconversion**

With image reject downconversion, your first reaction might be to just put a 90-degree phase shift in the RF path to the Q channel mixer, as shown in Figure 5.

**Figure 5**

**Initial approach to downconversion**

If you go through the math in a similar way as before, the IF frequency will be the difference between the RF and LO frequency. However, there are two drawbacks to this approach. The first is that the 90 degree phase shift in the RF path may be difficult to implement as this is a high frequency that is not fixed. The second problem is that it is still possible to have an image frequency at the input. Consider the case shown in Figure 6 where the intention is to convert a 5100 MHz signal down to 100 MHz using a 5000 MHz LO frequency. The problem is that an unwanted interferer signal at 4900 MHz could also downconvert to the same 100 MHz IF frequency. Although the approach in Figure 5 eliminates unwanted image frequencies from being produced at the output, it is still possible to have an image frequency (4900 MHz) at the input.

**Figure 6**

**Image Input Frequency Problem**

**Figure 7**

**Hartley image reject downconversion**

As the mathematics shows, the Hartley image reject down conversion approach as shown in Figure 7 solves both of the issues with the original approach in figure 5. There is no phase shift required in the RF path and an unwanted image frequency at the input is cancelled. For instance, if the RF frequency was intended to be 5100 MHz and the LO frequency was 5000 MHz, an unwanted interference frequency of 4900 MHz at the RF input would be cancelled and only the 5100 MHz signal would produce the desired 100 MHz IF frequency. Although this approach does require two low pass filters, these can be easily implemented with a few discrete components.

The approach in Figure 7 is often implemented with an analog-to-digital converter, which can perform the phase shifting in the digital domain.

**Figure 8**

**Downconversion with analog-to-digital converters**

**Reducing the mixer image using adjustable gain and phase**

Some devices, such as the LMX8410L from Texas Instruments, enable you to adjust the gain and phase to improve the mixer image. Although it may unrealistic to adjust these for every single device, you can correct a large amount of phase and gain imbalance if it follows a predictable trend in frequency.

Figure 9 shows the image response rejection ratio (IMRR) calculated with Equation 14 with and without correction. Without correction, it is closer to -28dBc; with correction, it is closer to -60dBc.

**Figure 9**

**LMX8410L image rejection calculated from gain and phase mismatch**

**Correlating gain/phase mismatch with a converted analog-to-digital signal**

As an exercise to see if Equation 14 really works, I hooked up the LMX8410L to the ADC32RF45EVM analog-to-digital converter board and measured the output signal to be about -29dBc with no correction as seen in Figure 10. So there is a reasonable match between the two approaches.

**Figure 10**

**Processed mixer image**

**Conclusion**

Converting frequencies up or down with a single mixer creates two sidebands: one desired and one undesired. You can eliminate the latter by either filtering or using two mixers in an image reject configuration.

**References**

- Sankar, Krishna. “Image Rejection Ratio (IMRR) with Transmit IQ Gain/Phase Imbalance.” DSPLOG, Jan. 31, 2013.
- Texas instruments LMX8410L data sheet.

Looking at all this math, I'm just glad that there is someone else who is able to help calculate all of that and make sure that the system is working properly. I look at it all and I'm worried about how much I could possibly screw up if any of my calculations are off. But also, perhaps that's the only way to improve – through trial and error to try and get your system working properly too right?