---
# System prepended metadata

title: Converters
tags: [' ss', Communication System Design, ' ncu']

---

---
tags: Communication System Design, ss, ncu
author: N0-Ball
title: Converters
GA: UA-208228992-1
---

# Intro

## Radio Receivers

- All radio receivers use Superhet principle
- Superhet receiver converts received signal to intermediate frequency (IF)

Superhet
:    supersonic heterodyne

Heterodyne
:    frequency multiplication

Supersonic
:    above audible hearing range

### TRF

:::info
a **sequence of RF amplifiers** and tuned narrow bandpass filters
:::

- Tuned filter must have $Q = \frac{f}{B} = 555$
    - **$f$** : frequency
    - **$B$** : bandwidth
- Almost impossible
    - Q for tuned RF filter is typically < 50

:::warning
FM radio at ~100 MHz, channel bandwidth is 180 kHz
:::

```graphviz
digraph {
  rankdir=LR;
  
  Antenna [shape="point", xlabel="Antenna"]
  RF_amp1 [shape="triangle", orientation=30, label="RF\namp"]
  BPF1 [shape="square", label="Tuned\nBPF"]
  RF_amp2 [shape="triangle", orientation=30, label="RF\namp"]
  BPF2 [shape="square", label="Tuned\nBPF"]
  dots [shape="none", label="..."]
  
  Antenna -> RF_amp1 -> BPF1 -> RF_amp2 -> BPF2 -> dots
}
```
**Antenna** -> RF amp -> Tuned BPF -> RF amp -> Tuned BPF ...

# Frequency Converter

Multiply input signal x(t) by local oscillator (LO) signal

- Shifts signal (to IF)

**LO output** 
- sine wave, s(t)
- frequency f~LO~
$$
s(t) = \cos (\omega_{LO}t)
$$

**Output of multiplier** $y'(t)$

$$
\begin{split}
y'(t) &= x(t) \times s(t) = A \cos(\omega_c t) \times \cos(\omega_{LO} t) \\[1em]
&= \frac{1}{2} A \cos((\omega_c + \omega_{LO})t) + \frac{1}{2} A \cos((\omega_c - \omega_{LO}) t) \\[1em]
&= \frac{1}{2} A \cos((\omega_c - \omega_{LO}) t) \quad \text{(BPF often selects lower side band)}
\end{split}
$$

## Image Frequencies

:::warning
There are always two RF frequencies that can be translated to the desired IF
:::

- Image frequency is always 2x IF from signal
- Need tuned RF filter to exclude image

![](https://i.imgur.com/OQMmRXP.png)

1. Signal -> Tuned RF BPF (Wanted Signal over LO left)
2. Signal Mix with LO (Signal with IF)
3. Signal -> IF BPF (IF Left)

### Example
:::info
Signal 89.1 Mhz with LO 10.7 MHz
:::

IF:
- $89.1 - 78.4 = 10.7$
- $78.4 - 67.7 = 10.7$

## Simplest case of x(t) = RF carrier

$$
x(t) = A \cos( \omega_c t)
$$

## Example #1
:::info
- Received signal at 89.1 MHz (carrier)
- Set LO frequency to 78.4 MHz
- BPF output for lower sideband at IF
    - $y(t) = \frac{1}{2} A \cos( (\omega_t - \omega_{LO}) t)$
:::

$$
f_{IF} = f_c - f_{LO} = 10.7\ mHz
$$

## Example #2
:::info
- Target signal change to 99.1MHz
- IF set to process 10.7 mHz
:::

$$
\begin{split}
&f_{IF} = f_c - f_{LO}\\[1em]
&\Rightarrow f_{LO} = f_c - f_{IF} = 88.4\ mHz
\end{split}
$$

:::warning
since IF BPF is fixed,
targeted frequency can be found by **adjusting LO frequency**
:::

# Narrowband Filter at Fixed IF Frequency

> 信號要搭飛機到目標
> 要上飛機(Modulation)
> 到站了要下飛機(Demodulation) [name=Cissi]

## Requires
:::info
Difficult to achieved with L-C filters
:::
- narrow bandwidth
- steeep sides
- flat top

## Compare

- RF BPF
    - broadband filter
    - reject image channel
- IF BPF
    - narrowband filter
    - reject adjacent channels

# Implementing Frequency Conversion

Any **non-linear circuit** will cause multiplication of input frequency

**For RF/Microwave**
- difficult
- expensive

## Simplest circuit - switch 

:::info
- Signal $v(t)$ applied to a switch
- Output: $y(t)$
:::

### Open

$$
y(t) = 0 \times v(t) = 0
$$

### Close

$$
y(t) = 1 \times v(t) = v(t)
$$

### Summary

$$
y(t) = s(t) \times v(t) \\[1em]
\left\{\begin{matrix}
s(t) = 0&,\ where\ nT \le t \lt \frac{T}{4} + nT \\[1em]
s(t) = 1&,\ where\ \frac{T}{4} + nT \le t \lt \frac{T}{2} + nT \\[1em]
\end{matrix}\right. ,\ 2n = m,\ m \in \Bbb Z \cup 0
$$

**Where**

$s(t)$
    : Switching Function
    
### Multiply

**Fourier series**
$$
s(t) = \frac{1}{2} + \frac{2}{\pi} \left[ \cos (\omega_0t) - \frac{1}{3} \cos (3 \omega_0t) + \cdots \right]
$$

**Output**
$$
y(t) = v(t) times s(t) = v(t) \times  + \left[ \frac{1}{2} + \frac{2}{\pi} \cos (\omega_0t) - \frac{2}{3\pi} \cos (3 \omega_0t) + \cdots \right]
$$

**BPF**
use BPF to select 
$$
y(t) = v(t) \times \frac{2}{\pi} \cos (\omega_0 t)
$$

**If $v(t) = A \cos(\omega t)$**
$$
\begin{split}
y(t) &=  A \cos(\omega t) \times \frac{2}{\pi} \cos (\omega_0 t) \\[1em]
&= \frac{2A}{\pi} \left[ \cos \left( (\omega - \omega_0) t \right) + \cos \left( (\omega + \omega_0) t \right) \right]
\end{split}
$$

## Diode 

![](https://wiki.analog.com/_media/university/courses/eps/eps_diode-curves-f2.png?w=500&tok=34583e)

- called mixer in radio receiver jargon
- used in up-convertors and down-convertors

## Others

- Balanced mixer (modulator)
- Double balanced mixer (DBM)
- Gain controlle amplifier

:::info
target: multiplication of input signal by a local oscilator frequency
:::

# Summary

- Almost all radio receivers use **superhat** design
- **Frequency conversion** is the key factor
    - multiplication (non-linear circuit)
    - select wih BPF after multiplication
- Incoming signal -> Intermediate Frequency
- IF has 
    - fixed frequency amplifier
    - narrow bandpass filters
    - = selectivity
- tuned RF bandpass filter removes image frequency noise