**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**

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):