Introduction to BPSK & BFSK Modulation Schemes and Shannon’s Theorem for Channel Capacity

Definition

BPSK (Binary Phase Shift Keying) is a digital modulation method in which the phase of a constant-frequency carrier wave is switched between two distinct values to represent binary 1 and 0, usually differing by 180 degrees.

BFSK (Binary Frequency Shift Keying) is a digital modulation method in which two different carrier frequencies are used to represent binary 1 and 0.

Shannon’s theorem for channel capacity states that the maximum reliable data rate of a communication channel depends on its bandwidth and signal-to-noise ratio (SNR), and is given by:

$$C = B \log_2(1 + \text{SNR})$$

where:

• $C$ = channel capacity in bits per second
• $B$ = bandwidth in hertz
• SNR = signal-to-noise ratio in linear form

Main Content

1. Binary Phase Shift Keying (BPSK)

Basic principle

In BPSK, the binary data is transmitted by changing the phase of a sinusoidal carrier signal. One bit value is assigned one phase, and the other bit value is assigned the opposite phase. Commonly, bit 1 may be represented by phase $0^\circ$ and bit 0 by phase $180^\circ$, or vice versa depending on the convention used.

Signal representation and behavior

The BPSK signal can be written as: $$s(t) = A_c \cos(2\pi f_c t + \phi)$$ where $\phi$ takes one of two values, typically $0$ or $\pi$. This means the waveform keeps the same amplitude and frequency, but the phase changes according to the input bits. Because the two phases are opposite, the constellation diagram consists of two points on the real axis at equal distance from the origin.

Example

Suppose the binary input is 1 0 1 1 0. If 1 is represented by phase $0^\circ$ and 0 by phase $180^\circ$, then the transmitted carrier alternates between these two phase states for each bit interval. This makes BPSK simple to implement and highly robust against noise compared with many other modulation techniques.

Advantages in communication

BPSK is very power efficient and has excellent noise immunity. It requires relatively low signal-to-noise ratio for correct detection, making it suitable for deep-space communication, wireless links, and systems where reliability is more important than very high data rate.

2. Binary Frequency Shift Keying (BFSK)

Basic principle

In BFSK, the binary data is represented by two different carrier frequencies. One frequency corresponds to binary 1, and the other frequency corresponds to binary 0. Unlike BPSK, the amplitude remains constant and the information is carried by frequency changes.

Signal representation and behavior

A BFSK system uses: $$s_1(t) = A_c \cos(2\pi f_1 t)$$ for one bit and $$s_0(t) = A_c \cos(2\pi f_0 t)$$ for the other bit, where $f_1 \neq f_0$. The separation between the frequencies must be sufficient so that the receiver can distinguish them properly. BFSK may be implemented as coherent BFSK or non-coherent BFSK, depending on whether the receiver uses phase information for detection.

Example

If binary 1 is assigned frequency 5 kHz and binary 0 is assigned frequency 2 kHz, then the transmitter sends a 5 kHz sinusoid for every 1 bit and a 2 kHz sinusoid for every 0 bit. A receiver detects which frequency is present during each bit interval and reconstructs the original data.

Advantages in communication

BFSK is robust and easy to implement, especially in noisy environments. It is less sensitive to amplitude variations than amplitude-based schemes and is often used in low-speed telemetry, early modems, and some radio communication systems.

Comparison with BPSK

BFSK generally requires more bandwidth than BPSK for the same bit rate, but it can be easier to detect in certain practical systems. BPSK usually provides better power efficiency, while BFSK may offer operational simplicity and acceptable reliability.

3. Shannon’s Theorem for Channel Capacity

Meaning of channel capacity

Shannon’s theorem gives the theoretical maximum data rate at which information can be transmitted over a channel with arbitrarily low probability of error, provided appropriate coding is used. It defines the upper limit, not the actual performance of a given system.

Mathematical form

The formula is: $$C = B \log_2(1 + \text{SNR})$$ where increasing bandwidth or improving SNR increases capacity. The logarithmic nature of the equation means that doubling SNR does not double capacity; the gain becomes smaller at high SNR values.

Interpretation of the theorem

• A wider bandwidth allows more signal changes per second, increasing capacity.
• A higher SNR means the signal is clearer relative to noise, allowing more reliable transmission.
• If the transmission rate is below $C$, reliable communication is theoretically possible.
• If the rate exceeds $C$, error-free communication cannot be achieved no matter how advanced the coding is.

Example

Suppose a channel has bandwidth $B = 3$ kHz and SNR = 15 (linear). Then: $$C = 3000 \log_2(1+15) = 3000 \log_2(16) = 3000 \times 4 = 12000 \text{ bps}$$ This means the maximum reliable transmission rate is 12 kbps under ideal coding conditions.

Relation to BPSK and BFSK

BPSK and BFSK are practical modulation schemes whose performance must be evaluated in light of Shannon’s limit. Even if a modulation scheme is efficient, it cannot exceed the channel capacity. BPSK typically achieves good performance at lower SNR, while BFSK may trade bandwidth for simplicity. Shannon’s theorem helps determine how close a system can theoretically come to error-free operation.

Working / Process

1. Convert binary data into modulation symbols

The transmitter first takes the incoming bit stream and maps each bit to a corresponding signal parameter. In BPSK, the bit is mapped to a phase value; in BFSK, it is mapped to a frequency value.

2. Transmit the modulated carrier through the channel

The selected carrier waveform is sent over the physical medium such as a wire, fiber, or wireless channel. During transmission, the signal is affected by attenuation, interference, and noise.

3. Detect, demodulate, and interpret the received signal

At the receiver, the waveform is analyzed to determine which phase or frequency was sent for each bit period. The recovered symbols are then translated back into binary data. In system design, the achievable rate and reliability are checked against Shannon’s capacity limit to ensure the communication is feasible.

Advantages / Applications

BPSK offers excellent noise performance and simple binary representation

• , making it a strong choice for low-SNR and power-limited communication systems.

BFSK provides robust communication with easy frequency-based detection

• , which is useful in low-complexity receivers and applications where amplitude fluctuations are severe.

Shannon’s theorem is essential for communication system design

• , because it sets the benchmark for maximum achievable data rate and helps engineers choose bandwidth, coding, and modulation strategies appropriately.

Summary

• BPSK sends binary data by changing the phase of a carrier wave.
• BFSK sends binary data by changing the frequency of a carrier wave.
• Shannon’s theorem gives the maximum reliable data rate of a noisy channel.

Important terms to remember

• BPSK, BFSK, carrier wave, phase, frequency, bandwidth, SNR, channel capacity, Shannon limit, modulation, demodulation