SM 5 BSZ - The complex mixer.
(June 3 2001)

#### The complex signal. I and Q.

We define a signal as an amplitude of some kind that varies with time. Signals may be electrical, accoustical or of some other kind. The instantaneous value of a physical quantity like the voltage across an antenna terminal or a microphone plug is a real quantity, a single signed number for each moment of time. This is a real valued signal V(t), a function that has a single value V (voltage) for each value of t (time).

It is well known that any function V(t) is identical to a sum of sine and cosine functions, the amplitudes of which constitute the Fourier transform.

It is thus perfectly acceptable to think of a signal as a sum of sine waves. When we do that we think about the signal in the frequency domain.

Thinking about signals in the frequency domain is something we do all the time like when saying a signal is transmitted in the 144MHz band. The complete description of a signal in the frequency domain is not just an amplitude for each frequency, there is a phase also. (the phase says how much is sine and how much is cosine)

To represent a signal in the frequency domain we need complex amplitudes. A complex number has a phase and an amplitude and it fits exactly to our needs.

Having found out that complex numbers are required for the frequency domain it is natural to use complex numbers for the signal itself, the signal in the time domain. What we see on an oscilloscope that is connected to a microphone or any other signal source is then only a part of the signal. The other part of the signal is missing!

To see the entire signal we need a dual trace oscilloscope. There is an infinite number of ways to make a voltage for the other trace. We may for example make I, the real part of the signal equal to the signal from the microphone while Q, the imaginary part is zero. If we do this and connect I and Q to a dual trace oscilloscope, of course, one track will show the microphone voltage while the other track is a straight line (zero volts). If we do it this way and make a complex fourier analysis we find that the spectrum is symmetric and that it contains equal amounts of positive and negative frequencies.

Another choice is to make I equal to the signal from the microphone while Q is a signal with the same amplitude but phase shifted by 90 degrees. If we do this and connect I and Q to a dual trace oscilloscope, both patterns are similar but certainly not equal. Complex signals (signal pairs) like this are present in SSB transmitters that use the phasing method. A complex fourier analysis will produce a spectrum that contains only positive or only negative frequencies depending on the direction of the 90 degree phase shift.

#### The real valued (ordinary) mixer

When two real valued signals are routed to a frequency mixer (an ideal one) the output will be the sum frequency and the difference frequency. This should be well known! In case we have a local oscillator at 116 MHz and connect a 28MHz transmitter to it we will get an output at 144MHz because 116+28=144 but we will also get an output at 88MHz because 116-28=88. When using ordinary mixers one has to use filters to remove undesired frequencies (good filters in this case. 88MHz may be the frequency where the neighbour a few meters away has his FM antenna while listening to FM broadcasting.)

The real mixer is a more or less ideal multiplier. The instantaneous output is the product of the two inputs. A typical schottky diode mixer like SBL-1 or SRA-1H is a four quadrant multiplier. It multiplies two signals with each other and can be used in many different ways.

If the two input signals are (t = 2 * PI * time in microseconds)

RF=sin(144*t)
LO=sin(116*t)

the output is IF=sin(144*t)*sin(116*t), the instantaneous product. From mathematics we learn that the product of two sine waves is equal to the sum of two sine waves, one of the sum frequency, the other of the difference frequency.

If the same signal is applied to two inputs, the output is the square of the input. It contains a DC component and the double frequency. This way the mixer can be used as a frequency doubler. In case the two signals have the same frequency but different phase the DC component will be proportional to the phase shift. The mixer can be used as a phase detector.

In case one signal is a DC voltage and the other is a RF signal the output is still the product. This means that the mixer can be used as a variable attenuator and/or 180 degree phase shifter. One has to use the IF port for the DC voltage and the other ones for input and output.

#### The complex mixer

The complex mixer makes the product of two complex signals. It produces a complex output signal. When one of the signals has a fixed frequency and constant amplitude, the complex mixer will produce either the sum frequency or the difference frequency depending on which way the 90 degree phase shift of the reference frequency is made. The complex mixer is used in SSB generation by the phasing method and it is also used in modern digital transmitters and receivers.

Writing complex numbers as pairs within a parenthesis the complex product is:

( A , B ) * ( C , D ) = ( A * C - B * D , A * D + B * C )

To realise the complex mixer in analog hardware one can use four real mixers. In case one signal is real and the other complex one may make A the real signal and B = 0 which causes two products to disappear An example of a real to complex mixer is the mixer pair used in direct conversion receivers.

In case one does not need the complex output, which is normally the case in phasing SSB transmitters, the complex to real mixer will produce one component only using a single mixer pair.

In digital processing, the full complex mixer can be used to shift the frequency of a signal without the need for any filtering. Digital processing is often more efficient with complex signals and conversion to the baseband I/Q pair is often done early in digital receivers, in the analog hardware or in a dedicated digital IC.

Shifting the frequency of a complex signal by use of a complex mixer in the time domain is exactly the same as shifting the frequency along the frequency axis (x-axis) in the frequency domain. All frequencies are increased or lowered by the same amount.

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