# Modulation roundup: error rates, noise, and capacity

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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,

.

Click here for Matlab simulation of bit error rate (BER) curve with BPSK modulation.

**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,

.

Click here for Matlab simulation of symbol error probability with 4PAM modulation

**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,

Click here for Matlab simulation of symbol error probability with 4QAM (QPSK) modulation

**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,

Click here for Matlab simulation of symbol error rate curve with 16QAM modulation

**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,

.

Click here for Matlab simulation of symbol error rate with 16PSK modulation

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,

Click here to download Matlab/Octave script for simulating symbol error rate for M-QAM modulation

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