BPSK, QPSK, 8PSK and QAM Calculator

CAVEAT: Please note that this calculator is calculating the Nyquist bandwidth. The Nyquist bandwidth is the minimum bandwidth than can be used to represent a signal. This is the correct bandwidth for transmitters which deploy a Nyquist bandwidth filter, which is the case for most professional transmitters.

If not filtered, the main lobe will have a bandwidth of twice the Nyquist bandwidth. For instance, a BPSK signal with an information rate of 1 Mbps will have a Nyquist bandwidth of 1 MHz. If unfiltered, the bandwidth of the main lobe is twice the Nyquist bandwidth or twice the symbol rate. In the case of a BPSK signal with an information rate of 1 Mbps this would be 2 MHz.

Technical Backgrounds

Any M-ary modulation scheme transmits some amount of bits per symbol. For BPSK only one bit is transfered for each symbol, two per symbol for QPSK and a whole 8 bits per symbol for 256 QAM.

Modulation Scheme	Bits per Symbol	Symbol Rate
BPSK	1	1 x bit rate
BFSK	1	1 x bit rate
QPSK	2	1/2 bit rate
8 PSK	3	1/3 bit rate
16 QAM	4	1/4 bit rate
32 QAM	5	1/5 bit rate
64 QAM	6	/6 bit rate
128 QAM	7	1/7 bit rate
256 QAM	8	1/8 bit rate

When transmitted, the symbol rate is equal to the 3 dB bandwidth of the transmitted signal. The 3 dB bandwidth is the frequency at which the signal amplitude is reduced by 3 dB. The occupied bandwith (OBW) refers to the bandwidth containing 99% of the total integrated power within the transmitted signal’s spectrum. The OBW is also oftentimes referred to as the channel bandwidth.

Forward error correction (FEC) repeats some parts of the data to provide redundancy. Implementing FEC allows the receiver to detect errors and correct the received signal without the need for retransmission. The downside of FEC is that the net throughput of data and thus the necessary bandwidth is increased proportionally to the degree of robustness implemented. The more robust a FEC scheme is, the larger the overhead. 1/2 FEC means that every data bit is transmitted twice. A FEC of 3/4 means that there’s 3 data bits per redundancy bits. Reed–Solomon codes are a more bandwidth-efficient way of providing error detection and correction. They also create some overhead, but not as much as traditional FEC.

Roll-off is the steepness of the filter-shape viewed in the frequency domain of the filter rthat is applied to the signal before transmission. As mentioned before, the unfiltered bandwidth of an M-ary PSK signal would ordinarily be twice the symbol rate. Therefore, root-raised-cosine (RRC) or square-root-raised-cosine (SRRC) filters are typically used in digital transmitting systems to limit the transmitted bandwidth to the part of the signal that is actually needed for reliable decoding (Nyquist bandwidth). These type of digital filters, however, can (and usually do) have filter-edges that roll-off slower than they theoretically could. A slower roll-off also causes the channel bandwidth to be slightly larger than required.