Modulation#
Introduction to Modulation#
Modulation in Communication Systems#
Digital data are typically represented as a stream of binary digits (0s and 1s).
These data can originate from:
A digital source (e.g., ASCII-encoded text from a computer terminal).
The analog-to-digital conversion of an analog source (e.g., digital audio or video).
The primary objective is to reliably transmit this data to the destination using the available communication channel.
Why Modulation is Needed?
To transmit the binary stream over a communication channel, we must generate a signal that:
Encodes the binary data, allowing retrieval of the original stream.
Matches the channel characteristics, ensuring its bandwidth aligns with the channel’s limitations.
Withstands impairments introduced by the channel, such as noise or interference.
This process of converting a digital sequence into a signal for transmission is known as digital modulation or digital signaling.
Modulation typically involves mapping a real-valued digital sequence (e.g., codewords) to a complex-valued signal using a specific modulation constellation. This lowpass-modulated signal is then shifted to a higher carrier frequency, to obtain a bandpass signal, for transmission. In this chapter, we focus on the mapping process in modulation.
Memoryless Modulation Schemes#
In modulation, the transmitted signals are typically bandpass signals, ensuring they fit within the bandwidth allocated by the communication channel.
The mapping between the digital sequence (assumed to be binary) and the transmitted signal sequence can be either memoryless or with memory, leading to memoryless modulation schemes and modulation schemes with memory.
In a memoryless modulation scheme, the binary sequence is divided into sub-sequences, bit-blocks, or messages, each of length \( k \). Each \( k \)-bit block is then mapped to one of the possible signals \( s_m (t) \), where \( 1 \leq m \leq 2^k \), without dependence on previously transmitted signals.
This scheme corresponds to a mapping from \( M = 2^k \) messages to \( M \) possible signals.
Mapping Messages to Symbols to Waveforms#
The standard process in digital communication systems where binary data is mapped to symbols and then to analog waveforms for transmission is as follows.
We map a message, which is a sequence of \(k\) binary bits, to a symbol represented by a point on a specific constellation diagram. Each symbol corresponds to a unique combination of bits and is depicted as a complex-valued point, indicating its amplitude and phase. This complex-valued symbol is then converted into an analog signal waveform, \(s_m(t), 1\leq m \leq M, M = 2^k\), which is transmitted over the communication channel.
Mapping Binary Bits to Symbols:
Message Definition: A message consists of a sequence of \( k \) binary bits (0s and 1s).
Symbol Representation: These \( k \) bits are grouped together and mapped to a symbol. Each symbol represents a unique combination of the \( k \) bits.
Constellation Diagram: The symbol is visually represented as a point on a constellation diagram. This diagram plots symbols in a two-dimensional plane, typically using the in-phase (I) and quadrature (Q) components, allowing for the visualization of amplitude and phase variations.
Symbol to Analog Signal Waveform:
Complex-Valued Symbol: The symbol, being a point on the constellation diagram, is inherently complex-valued, encompassing both amplitude and phase information.
Analog Mapping: This complex symbol is then converted into a time-domain analog signal waveform. This waveform is what gets transmitted over the physical communication channel (e.g., radio waves, optical fibers).
Definition: Signaling Interval
Each signal \( s_m(t) \in \mathbb{S} \) is transmitted every \( T_s \) seconds, where \( T_s \) [s] is known as the signaling interval.
This implies that in one second, \( R_s \) symbols are transmitted, where:
Definition: Signaling Rate
The parameter \( R_s \) is referred to as the signaling rate or symbol rate.
DISCUSSION
The term symbol used in this context aligns with its use in Symbol Error Rate (SER), a key performance metric in digital communication systems.
In digital modulation, a symbol represents a distinct waveform (or signal) that is transmitted within a given signaling interval \( T_s \). Each symbol corresponds to a group of bits (e.g., in QPSK, one symbol represents 2 bits; in 16-QAM, one symbol represents 4 bits).
The Symbol Error Rate (SER) measures the probability that a transmitted symbol is received incorrectly due to channel noise, interference, or distortion. It is defined as:
Since each symbol may represent multiple bits, SER is related to Bit Error Rate (BER), but BER accounts for individual bit errors, whereas SER considers errors at the symbol level. The relationship between SER and BER depends on the modulation scheme.
Thus, in our content, the symbol refers to the modulated signal unit, which is directly relevant when analyzing SER as a performance metric.
Definition: Bit Interval
Since each transmitted signal carries \( k \) bits of information, the bit interval \( T_b \) [s], the time duration in which one bit is transmitted, is given by:
Definition: Bit Rate
The bit rate \( R \) [bits/s], which represents the number of bits transmitted per second, is given by:
Energy of Waveforms#
Average Signal Energy#
If the energy content of a transmitted signal \( s_m(t) \) is denoted by \( \mathcal{E}_m \), then the average signal energy across all possible signals is given by:
where \( p_m \) represents the probability of transmitting the \( m \)-th signal (message probability).
Equiprobable Messages#
In the case of equiprobable messages (i.e., each message has an equal probability of being transmitted), the probability of each message is:
Thus, the average signal energy is given by:
Average Energy per Bit#
If all signals have the same energy, then \( \mathcal{E}_m = \mathcal{E} \), and consequently,
The average energy per bit—which represents the energy required for transmitting one bit of information—when signals are equiprobable is:
If all signals have equal energy \( \mathcal{E}_{sig} \), then the energy per bit simplifies to:
If a communication system transmits an average energy per bit of \( \mathcal{E}_{\text{bavg}} \) and each bit takes \( T_b \) seconds to be transmitted, then the average power sent by the transmitter is given by:
For the case where all signals have equal energy, this simplifies to:
SNR and \( \mathcal{E}_b/N_0 \)#
In an AWGN (Additive White Gaussian Noise) channel, the modulated signal is given by:
\[ x_{bp}(t) = \Re\{x_{lp}(t)e^{j2\pi f_c t}\} \]Before reception, this signal is corrupted by additive noise \( n_{bp}(t) \), which is a white Gaussian random process with zero mean and a power spectral density of \( N_0/2 \).
The received signal is thus:\[ r_{bp}(t) = x_{bp}(t) + n_{bp}(t) \]The signal-to-noise power ratio (SNR) at the receiver is defined as the ratio of the received signal power \( P_r \) to the noise power within the bandwidth of the transmitted signal \( x_{bp}(t) \).
The received power \( P_r \) depends on the transmitted power and various channel characteristics, including path loss, shadowing, and multipath fading.
The noise power is influenced by the bandwidth of the transmitted signal and the spectral properties of \( n_{bp}(t) \).
SNR and SINR#
If the bandwidth of the complex envelope \( x_{lp}(t) \) is \( B \), then the bandwidth of the transmitted signal \( x_{bp}(t) \) is \( 2B \).
Since the noise \( n_{bp}(t) \) has a uniform power spectral density of \( N_0/2 \), the total noise power within the bandwidth \( 2B \) is:
\[ N_{bp, tot} = \frac{N_0}{2} \times 2B = N_0B \]
Definition: Received SNR
The received signal-to-noise ratio (SNR) is given by:
where \( P_r \) is the received signal power, and \( N_0B \) represents the total noise power within the signal bandwidth.
In communication systems with interference, we often use the signal-to-interference-plus-noise ratio (SINR) instead of SNR for error probability calculations.
Definition: Received SINR
If the interference statistics approximate those of Gaussian noise, then using SINR as an alternative to SNR is a reasonable approximation. Let \( P_I \) be the average interference power; then the received SINR is given by:
SNR Expressions and Performance Metrics#
SNR in Terms of Energy per Bit and Energy per Symbol#
The signal-to-noise ratio (SNR) is often expressed using the signal energy per bit \( \mathcal{E}_b \) or per symbol \( \mathcal{E}_s \):
where:
\( T_s \) is the symbol duration (or signaling interval)
\( T_b \) is the bit duration (or bit interval)
For binary modulation, \( T_s = T_b \) and \( \mathcal{E}_s = \mathcal{E}_b \).
For data pulses where \( T_s = 1/B \) (e.g., raised cosine pulses with roll-off factor \( \beta = 1 \)), we obtain:
\( \text{SNR} = \frac{\mathcal{E}_s}{N_0} \) for multilevel signaling
\( \text{SNR} = \frac{\mathcal{E}_b}{N_0} \) for binary signaling
For general pulse shapes, \( T_s = k/B \) for some constant \( k \), leading to:
\[ k \cdot \text{SNR} = \frac{\mathcal{E}_s}{N_0} \]
SNR per Bit and Symbol Error Probability#
The quantities:
\( \gamma_s = \frac{\mathcal{E}_s}{N_0} \) (SNR per symbol)
\( \gamma_b = \frac{\mathcal{E}_b}{N_0} \) (SNR per bit)
are frequently used in system performance analysis.
For performance evaluation, we are primarily interested in the bit error probability \( P_b \) as a function of \( \gamma_b \).
However, in \( M \)-ary modulation schemes (e.g., M-PAM, M-PSK), the bit error probability depends on both:
The symbol error probability \( P_s \)
The symbol-to-bit mapping strategy
Hence, we typically compute the symbol error probability \( P_s \) as a function of \( \gamma_s \), using signal space concepts to analyze performance.