Digital Signal Analysis Problem 2 Fundamental Frequency And Fourier Series

In the realm of digital signal processing, understanding the characteristics of discrete-time signals is paramount. These signals, represented as sequences of values at specific time intervals, form the foundation for various applications, including audio processing, image analysis, and communication systems. Problem 2 delves into the analysis of a specific digital signal, offering a valuable opportunity to explore key concepts such as fundamental frequency, Fourier series coefficients, and signal sampling. In this article, we will embark on a comprehensive journey to dissect this problem, unraveling its intricacies and gaining a deeper understanding of the underlying principles.

The digital signal under consideration is defined as follows:

x(n) = 2cos(2π/3 * n) + 3sin(π/2 * n)

This signal is a combination of two sinusoidal components: a cosine wave with a frequency of 2π/3 and a sine wave with a frequency of π/2. The coefficients 2 and 3 represent the amplitudes of the respective sinusoidal components. Our objective is to analyze this signal in detail, addressing the following questions:

• What is the fundamental frequency for this signal?
• What are the Fourier series coefficients of x(n)?
• If x(n) is the sampled signal, what is the sampling frequency?

a) Determining the Fundamental Frequency

The fundamental frequency of a composite signal is the lowest frequency at which the signal repeats itself. In other words, it is the greatest common divisor (GCD) of the frequencies of the individual components. To determine the fundamental frequency of our signal, we need to find the GCD of 2π/3 and π/2. Let's denote the frequencies of the cosine and sine components as f1 and f2, respectively:

f1 = 2π/3
f2 = π/2

To find the GCD, we can first express the frequencies as fractions of π:

f1 = (2/3)π
f2 = (1/2)π

Now, we need to find the GCD of the fractions 2/3 and 1/2. To do this, we can find the least common multiple (LCM) of the denominators and then find the GCD of the numerators:

LCM(3, 2) = 6

Rewriting the fractions with the common denominator:

f1 = (4/6)π
f2 = (3/6)π

Now, we find the GCD of the numerators 4 and 3, which is 1. Therefore, the GCD of the fractions is 1/6.

Thus, the fundamental frequency (f0) of the signal is:

f0 = (1/6)π

This means that the signal x(n) repeats itself every 6 samples. This concept of fundamental frequency is crucial in signal processing as it helps in understanding the periodic nature of the signal and its spectral characteristics. The fundamental frequency provides a basis for analyzing the signal's harmonics, which are integer multiples of the fundamental frequency and contribute to the signal's overall shape and characteristics. In practical applications, knowing the fundamental frequency allows us to design appropriate filters and processing techniques for the signal.

b) Calculating Fourier Series Coefficients

The Fourier series is a mathematical tool that allows us to represent any periodic signal as a sum of sinusoidal components. These components are characterized by their frequencies, amplitudes, and phases. The Fourier series coefficients quantify the contribution of each sinusoidal component to the overall signal. For a discrete-time signal x(n) with a period of N, the Fourier series representation is given by:

x(n) = Σ[k=0 to N-1] a_k * exp(j * 2π/N * k * n)

where a_k are the Fourier series coefficients and j is the imaginary unit. To find the Fourier series coefficients, we use the following formula:

a_k = (1/N) * Σ[n=0 to N-1] x(n) * exp(-j * 2π/N * k * n)

In our case, the signal x(n) is given by:

x(n) = 2cos(2π/3 * n) + 3sin(π/2 * n)

and the period N is 12 (the least common multiple of 3 and 4, the denominators of the frequencies). To calculate the Fourier series coefficients, we can use Euler's formula to express the cosine and sine functions in terms of complex exponentials:

cos(θ) = (1/2) * (exp(jθ) + exp(-jθ))
sin(θ) = (1/2j) * (exp(jθ) - exp(-jθ))

Substituting these expressions into our signal equation, we get:

x(n) = 2 * (1/2) * (exp(j * 2π/3 * n) + exp(-j * 2π/3 * n)) + 3 * (1/2j) * (exp(j * π/2 * n) - exp(-j * π/2 * n))

Simplifying the equation:

x(n) = exp(j * 2π/3 * n) + exp(-j * 2π/3 * n) - (3j/2) * exp(j * π/2 * n) + (3j/2) * exp(-j * π/2 * n)

Now, we can identify the Fourier series coefficients by comparing this expression with the general Fourier series formula. We can see that the signal has four non-zero coefficients:

a_4 = 1  (corresponding to exp(j * 2π/3 * n))
a_8 = 1  (corresponding to exp(-j * 2π/3 * n))
a_3 = -3j/2 (corresponding to exp(j * π/2 * n))
a_9 = 3j/2  (corresponding to exp(-j * π/2 * n))

All other coefficients are zero. These coefficients represent the amplitudes and phases of the sinusoidal components that make up the signal. The Fourier series coefficients provide a frequency-domain representation of the signal, which is essential for various signal processing tasks such as filtering, spectral analysis, and signal reconstruction. Understanding the distribution of energy across different frequencies, as revealed by the Fourier series coefficients, is crucial for designing efficient signal processing algorithms.

c) Determining the Sampling Frequency

The concept of sampling frequency is critical when dealing with digital signals. Sampling is the process of converting a continuous-time signal into a discrete-time signal by taking measurements at regular intervals. The sampling frequency (fs) is the number of samples taken per unit of time, typically measured in Hertz (Hz). The choice of sampling frequency is governed by the Nyquist-Shannon sampling theorem, which states that to accurately reconstruct a signal, the sampling frequency must be at least twice the highest frequency component present in the signal.

In our case, the signal x(n) is a discrete-time signal, but we can imagine that it was obtained by sampling a continuous-time signal. To determine the appropriate sampling frequency, we need to identify the highest frequency component in the original continuous-time signal. The frequencies present in our discrete-time signal are 2π/3 and π/2. To relate these to continuous-time frequencies, we need to consider the relationship between the discrete-time frequency (ω) and the continuous-time frequency (f):

ω = 2πf/fs

where fs is the sampling frequency. Let's denote the continuous-time frequencies corresponding to 2π/3 and π/2 as f1 and f2, respectively:

2π/3 = 2πf1/fs  => f1 = fs/3
π/2 = 2πf2/fs   => f2 = fs/4

The highest frequency component is f1 = fs/3. According to the Nyquist-Shannon sampling theorem, the sampling frequency must be at least twice the highest frequency:

fs >= 2 * (fs/3)

This condition is always satisfied. However, to avoid aliasing, which is the distortion of the signal due to undersampling, we should choose a sampling frequency that is significantly higher than the Nyquist rate. A common rule of thumb is to choose a sampling frequency that is at least 2.5 to 3 times the highest frequency. In our case, this would suggest:

fs >= 3 * (fs/3) => fs >= fs

This doesn't give us a specific value for fs, but it indicates that any sampling frequency greater than fs/3 would theoretically work. However, in practice, we need to consider the specific application and the desired level of accuracy. A higher sampling frequency will result in a more accurate representation of the original signal but will also require more storage space and processing power. Therefore, choosing the appropriate sampling frequency involves a trade-off between accuracy and computational cost.

Problem 2 focuses on analyzing the digital signal x(n) = 2cos(2π/3 * n) + 3sin(π/2 * n). The key aspects to understand include determining the fundamental frequency of the signal, calculating its Fourier series coefficients, and considering the implications of sampling this signal. The questions to be addressed are: What is the fundamental frequency of x(n)? How can we determine the Fourier series coefficients for x(n)? If x(n) represents a sampled signal, what considerations apply to the sampling frequency?

Digital Signal Analysis Problem 2 Fundamental Frequency and Fourier Series