# Mixer math

```{contents} Contents
:class: this-will-duplicate-information-and-it-is-still-useful-here
```

## Up-conversion

![iq_mixer](../images/iq_mixer_symbol.plain.svg){align=center w=256px}

At the I and Q ports of the up-conversion mixer we have signals $I(t)$ and $Q(t)$ at frequency $\omega_S$:
```{math}
    I(t) &= X_I \cos(\omega_S t) - Y_I \sin(\omega_S t) \\
         &= A_I \cos(\omega_S t + \phi_I) \\
    Q(t) &= X_Q \cos(\omega_S t) - Y_Q \sin(\omega_S t) \\
         &= A_Q \cos(\omega_S t + \phi_Q)
```
where the following identities hold:
```{math}
:label: eye_apxy
    \begin{cases}
        A &= \sqrt{X^2 + Y^2} \\
        \phi &= \operatorname{atan2}(Y, X)
    \end{cases}
    \qquad
    \begin{cases}
        X &= A \cos{\phi} \\
        Y &= A \sin{\phi}
    \end{cases}
```

At the R port of the mixer, assuming a carrier frequency $\omega_N$ and zero phase, we get:
```{math}
    R(t) = I(t) \cos(\omega_N t) - Q(t) \sin(\omega_N t)
```
where the minus sign is a convention.
Expanding and rearranging, we get:
```{math}
:label: main_rf
    R(t) &= R_L(t) + R_U(t) \\
    R_L(t) &= \frac{1}{2}(X_I + Y_Q) \cos[(\omega_N -\omega_S)t] -\frac{1}{2}(X_Q - Y_I) \sin[(\omega_N -\omega_S)t] \\
    R_U(t) &= \frac{1}{2}(X_I - Y_Q) \cos[(\omega_N +\omega_S)t] -\frac{1}{2}(X_Q + Y_I) \sin[(\omega_N +\omega_S)t]
```
where $R_L$ and $R_U$ are the lower and upper sidebands, respectively.



### Zero intermediate frequency

When $\omega_S = 0$, at the I and Q ports of the mixer we get:
```{math}
    I(t) &= X_I = A_I \cos{\phi_I} \\
    Q(t) &= X_Q = A_Q \cos{\phi_Q}
```

At the R port:
```{math}
    R(t) &= X_I \cos(\omega_N t) -X_Q \sin(\omega_N t) \\
    &= A_I \cos{\phi_I} \cos(\omega_N t) -A_Q \cos{\phi_Q} \sin(\omega_N t)
```


#### Special case: same amplitude, same phase

Choosing $A_I = A_Q = A$ and $\phi_I = \phi_Q = \phi$ we get:
```{math}
    R(t) &= A \cos{\phi} \cos(\omega_N t) -A \cos{\phi} \sin(\omega_N t) \\
         &= \sqrt{2} A \cos{\phi} \cos(\omega_N t + \pi / 4 )
```
and a total amplitude $|R| = \sqrt{2}A|\cos{\phi}|$.
Note that the total phase is $\pi/4$, independent of $\phi$ (for $\phi \in [-\frac{\pi}{2},+\frac{\pi}{2}]$, otherwise there's a sign change).

Some notable cases:
| $\phi$  | $\|R\| \; / \; \sqrt{2}A$ |
| :----:  | :-------------------------------: |
| $0$     | $1$                               |
| $\pi/6$ | $\sqrt{3} / 2$                    |
| $\pi/4$ | $1 / \sqrt{2}$                    |
| $\pi/3$ | $1 / 2$                           |
| $\pi/2$ | $0$                               |


#### Special case: same amplitude, $\pi/2$ phase difference

Choosing $A_I = A_Q = A$, $\phi_I = \phi$ and $\phi_Q = \phi \mp \pi/2$ we get:
```{math}
    R(t) &= A \cos{\phi} \cos(\omega_N t)
    \mp A \sin{\phi} \sin(\omega_N t) \\
    &= A \cos(\omega_N \pm \phi)
```
and a total phase $\pm\phi$, and amplitude $|R| = A$ independent of $\phi$.
Note the inversion from $\mp$ to $\pm$.



### Single-sideband modulation (SSB)

:::{admonition} Quick reference
:class: tip
To perform single-sideband modulation, set {math}`\phi_I` to whatever you want and:
- {math}`\phi_Q = \phi_I + \pi/2` for lower sideband (LSB);
- {math}`\phi_Q = \phi_I - \pi/2` for upper sideband (USB).
:::


#### Lower sideband (LSB)

To get $R_U=0$ in Eq. {eq}`main_rf`, we want
```{math}
:label: lsb_sol
\begin{cases}
    X_I - Y_Q &= 0 \\
    X_Q + Y_I &= 0
\end{cases}
```
Then $I(t)$ and $Q(t)$ should have the same amplitude
```{math}
    A_I^2 = X_I^2 + Y_I^2 = Y_Q^2 + (-X_Q)^2 = A_Q^2
```
let's call the amplitude $A$.
We can rewrite Eq. {eq}`lsb_sol` as
```{math}
    \begin{cases}
        \cos{\phi_I} &= +\sin{\phi_Q} \\
        \cos{\phi_Q} &= -\sin{\phi_I}
    \end{cases}
```
which is solved by $\phi_Q = \phi_I + \pi/2$.

Substituting into $R_L(t)$ from Eq. {eq}`main_rf` we get:
```{math}
    R_L(t) &= X_I \cos[(\omega_N -\omega_S)t] + Y_I \sin[(\omega_N -\omega_S)t] \\
    &= A \cos{\phi} \cos[(\omega_N -\omega_S)t] + A \sin{\phi} \sin[(\omega_N -\omega_S)t] \\
    &= A \cos[(\omega_N -\omega_S)t - \phi ]
```


#### Upper sideband (USB)

Analogous to the previous section.
To get $R_L=0$ in Eq. {eq}`main_rf`, we want
```{math}
:label: usb_sol
\begin{cases}
    X_I + Y_Q &= 0 \\
    X_Q - Y_I &= 0
\end{cases}
```
Then $I(t)$ and $Q(t)$ should have the same amplitude
```{math}
    A_I^2 = X_I^2 + Y_I^2 = (-Y_Q)^2 + X_Q^2 = A_Q^2
```
let's call the amplitude $A$.
We can rewrite Eq. {eq}`usb_sol` as
```{math}
    \begin{cases}
        \cos{\phi_I} &= -\sin{\phi_Q} \\
        \cos{\phi_Q} &= +\sin{\phi_I}
    \end{cases}
```
which is solved by $\phi_Q = \phi_I - \pi/2$.

Substituting into $R_U(t)$ from Eq. {eq}`main_rf` we get:
```{math}
    R_U(t) &= X_I \cos[(\omega_N +\omega_S)t] -Y_I \sin[(\omega_N +\omega_S)t] \\
    &= A \cos{\phi} \cos[(\omega_N +\omega_S)t] -A \sin{\phi} \sin[(\omega_N +\omega_S)t] \\
    &= A \cos[(\omega_N +\omega_S)t + \phi ]
```




## Down-conversion



### General case

At the R port of the mixer, we have an incoming signal $R(t)$ at frequency $\omega_R$:
```{math}
    R(t) &= X_R \cos(\omega_R t) - Y_R \sin(\omega_R t) \\
         &= A_R \cos(\omega_R t + \phi_R)
```
where the identities of Eq. {eq}`eye_apxy` hold.

At the I and Q ports of the down-conversion mixer we get the signals $I(t)$ and $Q(t)$, assuming a
carrier frequency $\omega_N$ and zero phase:
```{math}
    I(t) &= +R(t) \cos(\omega_N t) \\
    Q(t) &= -R(t) \sin(\omega_N t)
```
where the minus sign is a convention. Expanding and rearranging, we get:
```{math}
    2I(t) &= X_R \cos[(\omega_R -\omega_N)t] - Y_R \sin[(\omega_R -\omega_N)t] \\
          &= A_R \cos[(\omega_R -\omega_N)t + \phi_R] \\
    2Q(t) &= Y_R \cos[(\omega_R -\omega_N)t] + X_R \sin[(\omega_R -\omega_N)t] \\
          &= A_R \cos[(\omega_R -\omega_N)t + \phi_R - \frac{\pi}{2}]
```
where we have omitted high-frequency components at frequency $\omega_R+\omega_N$. See the section
[](#spurs-in-down-conversion) below for a complete analysis of those spurious components.



### Zero intermediate frequency

In the special case when the incoming RF signal and down-conversion carrier have the same frequency
$\omega_R = \omega_N$, the I and Q ports of the mixer output the two quadratures of the original
baseband signal:
```{math}
    I(t) &= \frac{1}{2} X_R = \frac{1}{2} A_R \cos{\phi_R} \\
    Q(t) &= \frac{1}{2} Y_R = \frac{1}{2} A_R \sin{\phi_R}
```



### Single-sideband modulation (SSB)

The incoming RF signal $R(t)$ could contain a lower $R_L$ and an upper $R_U$ sideband:
```{math}
    R(t) &= R_L(t) + R_U(t) \\
    R_L(t) &= X_L \cos[(\omega_N -\omega_S)t] -Y_L \sin[(\omega_N -\omega_S)t] \\
    R_U(t) &= X_U \cos[(\omega_N +\omega_S)t] -Y_U \sin[(\omega_N +\omega_S)t]
```
where the sidebands are offset by $\omega_S$ from the central carrier at frequency $\omega_N$.
Typically we have that $\omega_S \ll \omega_N$.

After demodulation in the IQ mixer with a carrier at frequency $\omega_N$, we get at the I and Q
ports:
```{math}
:label: ssb_down
    2I(t) &= (X_L + X_U) \cos(\omega_S t) - (Y_U - Y_L) \sin(\omega_S t) \\
    2Q(t) &= (Y_L + Y_U) \cos(\omega_S t) - (X_L - X_U) \sin(\omega_S t)
```
where we have neglected high-frequency terms at $2\omega_N \pm \omega_S$.

Eq. {eq}`ssb_down` can be summarized in terms of the X and Y quadratures of the measured signals at
the I and Q ports:
```{math}
:label: ssb_down_quad
    \begin{cases}
    X_I &= \frac{1}{2} (X_L + X_U) \\
    Y_I &= \frac{1}{2} (Y_U - Y_L) \\
    X_Q &= \frac{1}{2} (Y_L + Y_U) \\
    Y_Q &= \frac{1}{2} (X_L - X_U)
    \end{cases}
```

Finally, we can invert Eq. {eq}`ssb_down_quad` to obtain the X and Y quadratures of the incoming RF signal:
```{math}
:label: ssb_down_quad_inv
    \begin{cases}
    X_L &= X_I + Y_Q \\
    Y_L &= X_Q - Y_I \\
    X_U &= X_I - Y_Q \\
    Y_U &= X_Q + Y_I
    \end{cases}
```
The utility function {func}`.utils.untwist_downconversion` uses the relations in Eq.
{eq}`ssb_down_quad_inv` to calculate the original upper and lower sidebands from the measured I and
Q signals.




## Spurs in down-conversion

We described in the previous section [](#down-conversion) how the down-conversion process generates
low-frequency components at frequency $\omega_R - \omega_N$, where $\omega_R$ is the frequency of
the incoming RF signal and $\omega_N$ is the frequency of the down-converting carrier generator.
These components are output from the I and Q ports, and we'll call them $I_{diff}(t)$ and
$Q_{diff}(t)$.

However, the down-conversion process generates also high-frequency components at frequency
$\omega_R + \omega_N$, which we'll call $I_{sum}(t)$ and $Q_{sum}(t)$ and can be viewed as spurious
signals. Overall, we have that $I(t) = I_{diff} + I_{sum}$ and $Q(t) = Q_{diff} + Q_{sum}$. In the
[previous section](#down-conversion) we neglected the high-frequency components, so we had assumed
$I(t) \approx I_{diff}$ and $Q(t) \approx Q_{diff}$.

Typically, we select a carrier frequency very close to the signal frequency, i.e.
$\omega_R-\omega_N \ll \omega_R+\omega_N$, so $I_{diff}$ and $I_{sum}$ are very far apart in
frequency space. In external, analog frequency down-conversion with IQ mixers and local oscillators
the terms $I_{sum}$ and $Q_{sum}$ at frequency $\omega_R+\omega_N$ fall outside the analog
bandwidth of the I and Q ports of the mixer and are heavily attenuated, so are usually negligible.
In Presto, however, the frequency down-conversion happens in the digital domain and therefore the
terms at $\omega_R+\omega_N$ are not attenuated: $I_{sum}$ and $Q_{sum}$ are still present in the
data, and are possibly aliased down to a lower frequency.

In this section, we will rewrite the [](#down-conversion) equations without neglecting the
high-frequency spurious terms.



### General case

At the R port of the mixer, we have an incoming signal $R(t)$ at frequency $\omega_R$:
```{math}
    R(t) &= X_R \cos(\omega_R t) - Y_R \sin(\omega_R t) \\
         &= A_R \cos(\omega_R t + \phi_R)
```

At the I and Q ports of the down-conversion mixer we get the signals $I(t)$ and $Q(t)$, assuming a
carrier frequency $\omega_N$ and zero phase:
```{math}
    I(t) &= +R(t) \cos(\omega_N t) \\
    Q(t) &= -R(t) \sin(\omega_N t)
```
where the minus sign is a convention. Expanding and rearranging, we get:
```{math}
    I(t) &= I_{diff}(t) + I_{sum}(t) \\
    2I_{diff}(t) &= X_R \cos[(\omega_R -\omega_N)t] - Y_R \sin[(\omega_R -\omega_N)t] \\
                 &= A_R \cos[(\omega_R -\omega_N)t + \phi_R] \\
    2I_{sum}(t)  &= X_R \cos[(\omega_R +\omega_N)t] - Y_R \sin[(\omega_R +\omega_N)t] \\
                 &= A_R \cos[(\omega_R +\omega_N)t + \phi_R]
```
Similarly, for $Q(t)$ we get:
```{math}
    Q(t) &= Q_{diff}(t) + Q_{sum}(t) \\
    2Q_{diff}(t) &= Y_R \cos[(\omega_R -\omega_N)t] + X_R \sin[(\omega_R -\omega_N)t] \\
                 &= A_R \cos[(\omega_R -\omega_N)t + \phi_R - \frac{\pi}{2}] \\
    2Q_{sum}(t)  &= -Y_R \cos[(\omega_R +\omega_N)t] - X_R \sin[(\omega_R +\omega_N)t] \\
                 &= A_R \cos[(\omega_R +\omega_N)t + \phi_R + \frac{\pi}{2}]
```



### Zero intermediate frequency

When $\omega_R = \omega_N$, at the I and Q ports of the mixer we get:
```{math}
    I(t) &= I_{diff}(t) + I_{sum}(t) \\
    2I_{diff}(t) &= X_R = A_R \cos(\phi_R) \\
    2I_{sum}(t)  &= X_R \cos(2 \omega_R t) - Y_R \sin(2 \omega_R t) \\
                &= A_R \cos(2 \omega_R t + \phi_R)
```
```{math}
    Q(t) &= Q_{diff}(t) + Q_{sum}(t) \\
    2Q_{diff}(t) &= Y_R = A_R \sin(\phi_R) \\
    2Q_{sum}(t)  &= Y_R \cos(2 \omega_R t) - X_R \sin(2 \omega_R t) \\
                &= A_R \cos(2 \omega_R t + \phi_R + \frac{\pi}{2})
```



### Single-sideband modulation (SSB)

The incoming RF signal $R(t)$ could contain a lower $R_L$ and an upper $R_U$ sideband:
```{math}
    R(t) &= R_L(t) + R_U(t) \\
    R_L(t) &= X_L \cos[(\omega_N -\omega_S)t] -Y_L \sin[(\omega_N -\omega_S)t] \\
    R_U(t) &= X_U \cos[(\omega_N +\omega_S)t] -Y_U \sin[(\omega_N +\omega_S)t]
```

After demodulation in the IQ mixer with a carrier at frequency $\omega_N$, we get at the I and Q
ports:
```{math}
    I(t) &= I_{diff}(t) + I_{sum}(t) \\
    2I_{diff}(t) &= (X_L + X_U) \cos(\omega_S t) - (Y_U - Y_L) \sin(\omega_S t) \\
    2I_{sum}(t)  &= X_L \cos[(2\omega_N-\omega_S) t] - Y_L \sin[(2\omega_N-\omega_S) t] \\
                 &+ X_U \cos[(2\omega_N+\omega_S) t] - Y_U \sin[(2\omega_N+\omega_S) t]
```
```{math}
    Q(t) &= Q_{diff}(t) + Q_{sum}(t) \\
    2Q_{diff}(t) &= (Y_L + Y_U) \cos(\omega_S t) - (X_L - X_U) \sin(\omega_S t) \\
    2Q_{sum}(t)  &= -Y_L \cos[(2\omega_N-\omega_S) t] - X_L \sin[(2\omega_N-\omega_S) t] \\
                 &+ -Y_U \cos[(2\omega_N+\omega_S) t] - X_U \sin[(2\omega_N+\omega_S) t]
```