# Modulation roundup: error rates, noise, and capacity

Krishna Pillai - July 06, 2008

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The article compares various digital modulation schemes like BPSK, QPSK, PAM, 16PSK, 32PSK, 16QAM and 64QAM using the following metrics:

(a) Symbol Error Rate vs. Signal to Noise Ratio (SER vs Es/No)

(b) Symbol Error Rate vs. Bit to Noise Ratio (SER vs Eb/No)

(c) Capacity in bits per second per Hertz vs. Bit to Noise Ratio (Capacity vs Eb/No)

(d) Bit Error Rate vs. Bit to Noise Ratio (BER vs Eb/No)

Symbol Error Rate vs. Es/No
Binary Phase Shift Keying (BPSK) Modulation
In Binary Phase Shift Keying, the symbols are used for transmitting information. From the post, Bit error probability for BPSK modulation, the symbol error rate is given as,

.

Pulse Amplitude Modulation (4-PAM)
In 4-PAM modulation, the symbols are used for transmitting information. The symbol error rate for 4-PAM modulation is derived in the post, symbol error rate for 4PAM and is given as,

.

4QAM (QPSK)
In 4-QAM modulation, the symbols are used for transmitting information. The symbol error rate for 4-QAM modulation, derived in the post, symbol error rate for 4-QAM (QPSK) is given as,

16QAM
In 16QAM modulation, the symbols are used. The symbol error rate for 16QAM modulation, derived in the post, symbol error rate for 16-QAM, is given as,

16PSK
In 16PSK modulation, the alphabets is used, where . The symbol error rate for 16PSK, derived in the post, Symbol Error Rate for 16PSK is given as,

.

Note: The formula derived in this post is for a general M-PSK case. For an M-PSK scheme, the symbol error rate is,

.

M-QAM
In a general M-QAM constellation, where and is even, the alphabets used are:

, where .

From the article deriving the symbol error rate for M-QAM,

Figure: Symbol Error Rate vs Es/No (dB) in AWGN

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

Symbol Error rate (SER) to Bit Error Rate (BER)
The information from the above table can be mapped into Shannon's capacity vs Eb/No curve.

Figure: Shannon's capacity curve for various digital modulation schemes.

Note: The Figure 10.44 in [COMMUNICATION SYSTEMS: PROAKIS, SALEHI], we can see that the points 2PAM (SSB) and QPSK are overlapping; the points 4PAM (SBB) and 16QAM are overlapping. This implies that the SNR per bit (Eb/No) required for achieving symbol error rate of is the same for 2PAM and QPSK; and 4PAM and 16QAM respectively. However, from the symbol error rate vs Eb/No plot, we know that for the same value of SNR per bit (Eb/No), the symbol error rate for QPSK is double that of BPSK; and the symbol error rate of 16QAM is double that of 4PAM. So, I think that the points should not overlap (should be offset by around 0.3dB). I will update once I get a response from the authors.

Symbol Error rate (SER) to Bit Error Rate (BER)
Typical communication systems use Gray coded modulation mapping, i.e bits represented in adjacent symbols differ by bit only. When a symbol is incorrectly decoded, it typically falls into the adjacent the symbol bin. Hence, each symbol error causes one bit out of bits to be in error.

So, the relation between symbol error and bit error is,

.

With this approximation, the bit error rate equations are:

Figure: BER vs SNR per bit (Eb/No) for digital modulation schemes

References
Fundamentals of Communication Systems, by John G. Proakis, Masoud Salehi