Constellation Diagrams: ASK, FSK, and PSK
Comprehensive guide to signal space representation, including interactive simulators and MATLAB implementations.
BASK (Binary ASK) Modulation
Transmits one of two signals: 0 or -√Eb, where Eb is the energy per bit. These signals represent binary 0 and 1.
BFSK (Binary FSK) Modulation
Transmits one of two signals: +√Eb (On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset) or √Eb (on x-axis), where Eb is the energy per bit. These signals represent binary 0 and 1.
BPSK (Binary PSK) Modulation
Transmits one of two signals: +√Eb or -√Eb (they differ by 180 degree phase shift), where Eb is the energy per bit. These signals represent binary 0 and 1.
Signal Space Simulator
Visualize BASK, BPSK, and BFSK Constellation Diagrams with Noise Control.
Theory & Key Points
- ✔ BASK: Bit '0' is low voltage/no signal; bit '1' is high level voltage.
- ✔ BFSK: Maps bit '0' to 'j' and bit '1' to '1'. Signals are orthogonal.
- ✔ BPSK: 0° shift for binary '1' (+1) and 180° shift for binary '0' (-1). Read more [↗]
Figure 1: Constellation diagrams of ASK, PSK, and FSK. The x-axis shows the real part, y-axis shows the imaginary part.
The spacing between signaling points determines the Probability of Error (Pe). ASK is prone to bit errors due to shorter distances ($\sqrt{E_b}$). PSK performs better in noisy channels with a distance of $2\sqrt{E_b}$.
MATLAB Code Implementation
ASK vs FSK vs PSK: Comparison Table
| Parameter | BASK | BFSK | BPSK |
|---|---|---|---|
| Abbreviation | Amplitude Shift Keying | Frequency Shift Keying | Phase Shift Keying |
| Noise Immunity | Very Low | High | Highest |
| Bit Rate | Lower | Moderate | Higher |
Signal Representation Formulas
Mathematically, the transmitted signals for binary modulations can be expressed as:
BPSK Equation:
$$s(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t + \theta)$$where theta is 0 or pi radians.
BFSK Equation:
$$s(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_i t)$$where f_i is the mark or space frequency. (different frequencies for bit 0 or 1).
BFSK Constellation: Orthogonal vs. Antipodal Analysis
In signal space theory, the "distance" between signaling points determines the error performance. Let's compare Orthogonal BFSK with the Antipodal benchmark.
📐 Orthogonal BFSK
Used when frequencies are separated by $1/2T_b$. The symbols are perpendicular.
$s_1 = (\sqrt{E_b}, 0)$
$s_2 = (0, \sqrt{E_b})$
↔️ Antipodal (BPSK)
The "Gold Standard" of power efficiency where signals are mirror images.
$s_1 = +\sqrt{E_b}$
$s_2 = -\sqrt{E_b}$
"Because the distance between points in Antipodal signaling ($$2\sqrt{E_b}$$) is greater than in Orthogonal BFSK ($$\sqrt{2E_b}$$), BPSK requires approximately 3dB less power to achieve the same Bit Error Rate (BER) as BFSK."
Signal Space & The Q-Function
An interactive guide to understanding modulation boundaries and error probabilities.
Signal Space Simulator
Watch symbols cross the Decision Boundary (Red Line) as noise increases.
The Q-Function: Tail Probability
The Q-function $Q(x)$ tells us the probability that a Gaussian noise sample exceeds a certain distance $x$. This is the mathematical "Red Area" of errors.
Modulation Comparison Table
| Modulation | Distance ($d$) | BER Formula | Efficiency |
|---|---|---|---|
| BPSK | $2\sqrt{E_b}$ | $Q(\sqrt{2E_b/N_0})$ | Highest (Best) |
| BFSK | $\sqrt{2E_b}$ | $Q(\sqrt{E_b/N_0})$ | Moderate (-3dB) |
| BASK | $\sqrt{E_b}$ | $Q(\sqrt{E_b/2N_0})$ | Lowest |
How to Calculate Normalized Distance ($x$)
In digital communications, the argument $x$ inside the $Q(x)$ function represents the ratio of the signal's "safety margin" to the noise magnitude. In an AWGN channel, the noise standard deviation is defined as: $\sigma = \sqrt{N_0 / 2}$
BPSK
Distance to boundary is $\sqrt{E_b}$
$x = \sqrt{\frac{2E_b}{N_0}}$
BFSK
Distance to boundary is $\frac{\sqrt{E_b}}{\sqrt{2}}$
$x = \sqrt{\frac{E_b}{N_0}}$
BASK
Distance to boundary is $\frac{\sqrt{E_b}}{2}$
$x = \sqrt{\frac{E_b}{2N_0}}$
The 3dB BPSK Advantage in compare to BFSK
The performance of a modulation scheme is directly proportional to the Euclidean Distance between signaling points. Let's derive why BPSK outperforms BFSK by exactly 3dB.
Step 1: Distance in BPSK
BPSK uses Antipodal signaling. The distance between $+\sqrt{E_b}$ and $-\sqrt{E_b}$ is:
Step 2: Distance in BFSK
Orthogonal BFSK points are perpendicular. Using the Pythagorean theorem:
The 3dB Conclusion
Taking the ratio of the squared distances:
BPSK is 3dB more power-efficient than BFSK. In practical terms, BPSK requires half the transmitter power to achieve the same reliability as BFSK.
ASK Use Case
Primarily used in Optical Fiber communications and simple IR remote controls.
FSK Use Case
Widely used in Caller ID systems and low-frequency radio modems.
PSK Use Case
Used in Bluetooth, Wi-Fi, and Satellite communications (GPS).
Frequently Asked Questions
What is the main advantage of BPSK over BFSK? ▼
BPSK is more bandwidth-efficient and has a lower Bit Error Rate (BER) for the same Signal-to-Noise Ratio (SNR) compared to BFSK.
Why is noise added to the constellation diagram? ▼
In real-world channels, AWGN (Additive White Gaussian Noise) causes the signaling points to "spread" into clusters. If the noise is high enough, these clusters overlap, leading to decision errors at the receiver.