# Modulation roundup: error rates, noise, and capacity

**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.

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