Let's assume your original digital message bitstream is: 0, 0, 1, 0, 0, 0, 1, 0, 1, 1
In 4-QAM, we group them into pairs: (00), (10), (00), (10), (11). Your baseband symbols are:
- Symbol 1 (Bits 00): -1.00 - j1.00
- Symbol 2 (Bits 10): 1.00 - j1.00
- Symbol 3 (Bits 00): -1.00 - j1.00
- Symbol 4 (Bits 10): 1.00 - j1.00
- Symbol 5 (Bits 11): 1.00 + j1.00
To transmit these symbols over a wireless medium, we modulate this baseband signal onto a high-frequency carrier (e.g., 50 Hz). This process creates the passband signal, where the information is stored in the phase and amplitude of the sine wave.
Fig 1: 4-QAM Baseband I and Q Components
Fig 2: 4-QAM Passband Modulated Signal
In this example, the symbol rate is 5 symbols per second.
Detailed Explanation
4-QAM Constellation Mapping
In standard 4-QAM mapping, bits are converted to complex points on a grid:
| Bits | Symbol (I + jQ) | Phase (Angle) |
|---|---|---|
| 00 | -1 - j1 | -135° |
| 01 | -1 + j1 | 135° |
| 11 | +1 + j1 | 45° |
| 10 | +1 - j1 | -45° |
Magnitude of a Complex Number
For any complex number z = I + jQ, the magnitude (amplitude) is:
|z| = √(I² + Q²)
For a 4-QAM symbol using coordinates (1,1), the magnitude is √(1² + 1²) = √2 ≈ 1.414. All four constellation points share this same magnitude, which is why 4-QAM is sometimes called 4-PSK.
How We Compute Phase
The phase angle Ï• is found using the arctangent of the Quadrature (Q) and In-phase (I) components:
arg(1 + j1) = tan⁻¹(1/1) = 45°arg(−1 + j1) = tan⁻¹(1/−1) = 135°arg(−1 − j1) = tan⁻¹(−1/−1) = −135°arg(1 − j1) = tan⁻¹(−1/1) = −45°