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Design Article

# Modulation roundup: error rates, noise, and capacity

## 7/6/2008 12:00 PM EDT

Symbol error rate vs Eb/No
Symbol error rate vs Eb/No
The relation between bit energy Eb/No and symbol energy Es/No is reasonably straight forward. For M-PSK/M-QAM modulation, the number bits in each constellation symbol is,

Since each symbol carries bits, the symbol to noise ratio is times the bit to noise ratio , ie.

.

Plugging in the above formula, the symbol error rate vs bit energy (SNR per bit, Eb/No) is given as,

Figure: Symbol Error Rate vs SNR per bit (Eb/No) for digital modulation schemes

Bandwidth requirements and Capacity
From the post, Transmit pulse shaping filter, we know that minimum required bandwidth for transmitting symbols with symbol period without causing inter symbol interference (ISI) is Hz.

Further, if the transmission is passband, PAM transmission requires bandwidth of Hz (Refer to post on Need for IQ modulator and demodulator). However, the spectral efficiency can be improved by either,

(a) Filtering the unwanted half of the bandwidth from the passband PAM, resulting in a bandwidth requirement of Hz— called single sideband modulation (SSB).

(b) Using both I and Q arm for modulation, resulting in a bandwidth requirement of Hz— called QAM (quadrature amplitude modulation).

Based on knowledge of symbol duration and bandwidth requirement, the capacity in bits per second per Hz for various modulation schemes can be derived. For example, for 16QAM modulation with symbol duration , the bit rate is bits per second (as each symbol carries 4 bits) and the bandwidth required is Hz.

Further, from the Symbol Error rate vs Eb/No plot, the Bit to Noise ratio (Eb/No) required for achieving arbitrarily low symbol error probability of can be obtained.

.

Table: Bandwidth, Capacity and Eb/No requirements for symbol error rate of 10^-5

wassouf_salahat

12/22/2008 5:51 PM EST

thatsd great keep going:)

tamdtvt

5/12/2009 1:08 PM EDT

please give me sourse code ??

DerekJC

9/4/2009 6:18 AM EDT

Good article. However, it would be useful to include some constellation plots.

dangermouse

9/16/2011 5:11 PM EDT

So if I'm setting up a RF Central Microwave system for a reality TV show and my options are QPSK, 16QAM, or 64QAM. What should I set the transmitter to, to obtain the best quality in the microwave monitor/receiver? And does anyone know about the FEC: 1/2 2/3 3/4 5/6 7/8
Guard interval: 1/32 1/16 1/8 1/4
What are my optimum settings?

Frank Eory

9/16/2011 6:13 PM EDT

The absolute most robust (error-free) settings would be QPSK, rate 1/2 FEC and 1/4 guard interval -- but that might not give you enough Mbps data capacity for what you're doing.

Step one is figure out the data rate you need to transmit. Step two, dial in the most robust parameters that are just enough to meet that data rate. Step three, cross your fingers and hope there is enough signal to noise ratio for the receiver to catch all the data with an extremely low bit error rate.

banibani

4/25/2012 12:19 PM EDT

plz help me by giving me the formula of 16-APSK bit error rate or symbol error rate.
i need only the equation of APSK bit error rate or symbol error rate.