Tag: Derivation

Analog and Discrete signals

In the context of signal and systems, analog and discrete signals are two different types of signals that convey information.

Analog signal

An analog signal is a continuous signal that varies smoothly over time. It can take on any value within a certain range. Analog signals are represented by physical quantities such as voltage, current, or sound waves. For example, the varying voltage produced by a microphone when recording sound is an analog signal. Analog signals are typically represented as continuous waveforms.

Let’s consider an example of a simple analog signal:

\[x(t) = A \cdot sin \left(2 \pi f t + \phi \right)\]

    In this equation:

    • x(t) represents the value of the analog signal at time t.
    • A is the amplitude of the signal, which determines its maximum value.
    • f is the frequency of the signal, which represents the number of cycles per unit of time.
    • φ is the phase of the signal, which represents the offset or starting point of the waveform.

    This equation describes a sinusoidal analog signal, where the value of the signal varies continuously over time. The signal can have an infinite number of values at any given instant.

    Discrete signal

    On the other hand, a discrete signal is a signal that is defined only at specific instances of time and takes on a finite set of values. Discrete signals are often derived from analog signals by a process called sampling, where the continuous analog signal is measured or sampled at regular intervals. Each sample represents the value of the signal at a particular instant. These samples can be stored and processed using digital systems. Examples of discrete signals include digital audio, digital images, and the output of a digital sensor.

    Discrete signals are commonly used in digital signal processing and can be represented using mathematical equations.

    The general equation for a discrete signal can be written as:

    \[x[n] = f(n)\]

    In this equation:

    • x[n] represents the value of the discrete signal at time instance n.
    • f(n) is the function that determines the value of the signal at each specific time instance.

    The function f(n) can take various forms depending on the specific characteristics of the discrete signal. For example, let’s start with the equation for the analog sinusoidal signal:

    \[x(t) = A \cdot sin \left(2 \pi f t + \phi \right)\]

    To obtain the discrete version of this signal, we need to sample it at regular intervals. The sampling process involves measuring the analog signal at equidistant points in time.

    Let’s define the sampling period as \(T_s\), which represents the time between two consecutive samples. The sampling rate is the inverse of the sampling period and is denoted as \(f_s = 1 / T_s\).

    Now, we can express the discrete version of the sinusoidal signal as:

    \[x[n] = x(n T_s) = A \cdot sin(2 \pi f n T_s + \phi)\]

    In this equation:

    • x[n] represents the value of the discrete signal at sample index n.
    • f is the frequency of the sinusoidal signal in hertz
    • n represents the sample index, indicating which sample we are considering.
    • \(T_s\) is the sampling period.
    • \(f_s\) is the sampling frequency, which is the reciprocal of the sampling period.

    By substituting \(nT_s\) for t in the analog sinusoidal signal equation, we obtain the discrete version of the sinusoidal signal. The discrete signal represents the sampled values of the original analog signal at each specific time instance, \(nT_s\).

    It’s important to note that the accuracy of the discrete signal representation depends on the sampling rate. According to the Nyquist-Shannon sampling theorem, for real signals, the sampling rate should be at least twice the maximum frequency of the analog signal to avoid aliasing and accurately reconstruct the signal from its samples.

    Python code

    Following is an example Python code that simulates an analog sinusoidal signal, samples it to obtain a discrete version, and overlays the two signals for comparison

    import numpy as np
import matplotlib.pyplot as plt

# Parameters for the analog signal
amplitude = 1.0  # Amplitude of the signal
frequency = 2.0  # Frequency of the signal in Hz
phase = 0.0  # Phase of the signal in radians

# Parameters for the discrete signal
sampling_rate = 10  # Number of samples per second
num_samples = 20

# Time arrays for the analog and discrete signals
t_analog = np.linspace(0, num_samples / sampling_rate, num_samples * 10)  # Higher resolution for analog signal
n_discrete = np.arange(num_samples)

# Generate the analog signal
analog_signal = amplitude * np.sin(2 * np.pi * frequency * t_analog + phase)

# Sample the analog signal to obtain the discrete signal
discrete_signal = amplitude * np.sin(2 * np.pi * frequency * n_discrete / sampling_rate + phase)

# Plot the analog and discrete signals
plt.plot(t_analog, analog_signal, label='Analog Signal')
plt.stem(n_discrete / sampling_rate, discrete_signal, 'r', markerfmt='ro', basefmt=' ', label='Discrete Signal')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Analog and Discrete Sinusoidal Signals')
# Move the legend outside the figure
plt.legend(loc='upper right', bbox_to_anchor=(1.1, 1))
plt.grid(True)
plt.show()

    Resulting plot

    Simulated analog and discrete sinusoidal signals
    Figure 1: Simulated analog and discrete sinusoidal signals

    Derive BPSK BER – optimum receiver in AWGN channel

    Key focus: Derive BPSK BER (bit error rate) for optimum receiver in AWGN channel. Explained intuitively step by step.

    BPSK modulation is the simplest of all the M-PSK techniques. An insight into the derivation of error rate performance of an optimum BPSK receiver is essential as it serves as a stepping stone to understand the derivation for other comparatively complex techniques like QPSK,8-PSK etc..

    Understanding the concept of Q function and error function is a pre-requisite for this section of article.

    The ideal constellation diagram of a BPSK transmission (Figure 1) contains two constellation points located equidistant from the origin. Each constellation point is located at a distance from the origin, where Es is the BPSK symbol energy. Since the number of bits in a BPSK symbol is always one, the notations – symbol energy (Es) and bit energy (Eb) can be used interchangeably (Es=Eb).

    Assume that the BPSK symbols are transmitted through an AWGN channel characterized by variance = N0/2 Watts. When 0 is transmitted, the received symbol is represented by a Gaussian random variable ‘r‘ with mean=S0 = and variance =N0/2. When 1 is transmitted, the received symbol is represented by a Gaussian random variable – r with mean=S1= and variance =N0/2. Hence the conditional density function of the BPSK symbol (Figure 2) is given by,

    Figure 1: BPSK – ideal constellation
    Figure 2: Probability density function (PDF) for BPSK Symbols

     An optimum receiver for BPSK can be implemented using a correlation receiver or a matched filter receiver (Figure 3). Both these forms of implementations contain a decision making block that decides upon the bit/symbol that was transmitted based on the observed bits/symbols at its input.

    Figure 3: Optimum Receiver for BPSK

    When the BPSK symbols are transmitted over an AWGN channel, the symbols appears smeared/distorted in the constellation depending on the SNR condition of the channel. A matched filter or that was previously used to construct the BPSK symbols at the transmitter. This process of projection is illustrated in Figure 4. Since the assumed channel is of Gaussian nature, the continuous density function of the projected bits will follow a Gaussian distribution. This is illustrated in Figure 5.

    Figure 4: Role of correlation/Matched Filter

    After the signal points are projected on the basis function axis, a decision maker/comparator acts on those projected bits and decides on the fate of those bits based on the threshold set. For a BPSK receiver, if the a-prior probabilities of transmitted 0’s and 1’s are equal (P=0.5), then the decision boundary or threshold will pass through the origin. If the apriori probabilities are not equal, then the optimum threshold boundary will shift away from the origin.

    Figure 5: Distribution of received symbols

    Considering a binary symmetric channel, where the apriori probabilities of 0’s and 1’s are equal, the decision threshold can be conveniently set to T=0. The comparator, decides whether the projected symbols are falling in region A or region B (see Figure 4). If the symbols fall in region A, then it will decide that 1 was transmitted. It they fall in region B, the decision will be in favor of ‘0’.

    For deriving the performance of the receiver, the decision process made by the comparator is applied to the underlying distribution model (Figure 5). The symbols projected on the axis will follow a Gaussian distribution. The threshold for decision is set to T=0. A received bit is in error, if the transmitted bit is ‘0’ & the decision output is ‘1’ and if the transmitted bit is ‘1’ & the decision output is ‘0’.

    This is expressed in terms of probability of error as,


    Or equivalently,

    By applying Bayes Theorem↗, the above equation is expressed in terms of conditional probabilities as given below,


    Since a-prior probabilities are equal P(0T)= P(1T) =0.5, the equation can be re-written as

    Intuitively, the integrals represent the area of shaded curves as shown in Figure 6. From the previous article, we know that the area of the shaded region is given by Q function.

    Figure 6a, 6b: Calculating Error Probability

    Similarly,

    From (4), (6), (7) and (8),


    For BPSK, since Es=Eb, the probability of symbol error (Ps) and the probability of bit error (Pb) are same. Therefore, expressing the Ps and Pb in terms of Q function and also in terms of complementary error function :


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    Reference

    [1] Nguyen & Shwedyk, “A First course in Digital Communications”, Cambridge University Press, 1st edition.↗

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