Properties and Application of Discrete Time Fourier Series

Introduction

The discrete time Fourier series (DTFS) is a mathematical tool used in the field of Signals & Systems to analyze and manipulate discrete-time signals. It is an extension of the continuous-time Fourier series, which is used for continuous-time signals. The DTFS allows us to represent a periodic discrete-time signal as a sum of complex exponential functions. This representation provides insights into the frequency content and other properties of the signal.

Importance of Discrete Time Fourier Series in Signals & Systems

The DTFS is an essential tool in Signals & Systems for several reasons:

1. It allows us to analyze the frequency content of a discrete-time signal, which is crucial for understanding its behavior and characteristics.
2. It provides a mathematical framework for signal processing techniques such as filtering, modulation, and spectrum analysis.
3. It enables us to represent periodic signals in a concise and computationally efficient manner.

Fundamentals of Discrete Time Fourier Series

Before diving into the properties and applications of the DTFS, let's briefly review its fundamentals.

The DTFS representation of a periodic discrete-time signal x[n] is given by:

$$x[n] = \sum_{k=0}^{N-1} X[k]e^{j\frac{2\pi}{N}kn}$$

where:

• x[n] is the periodic discrete-time signal
• N is the period of the signal
• X[k] is the complex amplitude of the k-th harmonic component
• j is the imaginary unit

The DTFS coefficients X[k] can be obtained using the Discrete Fourier Transform (DFT) algorithm.

Properties of Discrete Time Fourier Series

The DTFS possesses several important properties that allow us to manipulate and analyze signals effectively. Let's explore these properties in detail.

Linearity Property

The linearity property of the DTFS states that the DTFS of a linear combination of signals is equal to the linear combination of their individual DTFS representations.

Explanation of Linearity Property

The linearity property can be understood as follows: if we have two discrete-time signals x1[n] and x2[n] with DTFS coefficients X1[k] and X2[k] respectively, and we form a new signal x[n] by taking a linear combination of x1[n] and x2[n] as:

$$x[n] = a \cdot x1[n] + b \cdot x2[n]$$

where a and b are constants, then the DTFS coefficients of x[n] can be expressed as:

$$X[k] = a \cdot X1[k] + b \cdot X2[k]$$

Mathematical Representation of Linearity Property

The mathematical representation of the linearity property is given by:

$$\text{DTFS}(a \cdot x1[n] + b \cdot x2[n]) = a \cdot \text{DTFS}(x1[n]) + b \cdot \text{DTFS}(x2[n])$$

Example Illustrating Linearity Property

Let's consider two discrete-time signals:

$$x1[n] = [1, 2, 3, 4]$$ $$x2[n] = [5, 6, 7, 8]$$

The DTFS coefficients of x1[n] and x2[n] are given by:

$$X1[k] = [10, -2j, -2, -2j]$$ $$X2[k] = [26, -6j, -6, -6j]$$

Now, let's form a new signal x[n] by taking a linear combination of x1[n] and x2[n] with a = 0.5 and b = 0.3:

$$x[n] = 0.5 \cdot x1[n] + 0.3 \cdot x2[n]$$

The DTFS coefficients of x[n] can be calculated as:

$$X[k] = 0.5 \cdot X1[k] + 0.3 \cdot X2[k]$$

Time Shifting Property

The time shifting property of the DTFS states that shifting a discrete-time signal in the time domain corresponds to multiplying its DTFS coefficients by a complex exponential function.

Explanation of Time Shifting Property

The time shifting property can be understood as follows: if we have a discrete-time signal x[n] with DTFS coefficients X[k], and we shift x[n] by M samples to obtain a new signal y[n] as:

$$y[n] = x[n - M]$$

then the DTFS coefficients of y[n] can be expressed as:

$$Y[k] = X[k] \cdot e^{-j\frac{2\pi}{N}kM}$$

where N is the period of the signal.

Mathematical Representation of Time Shifting Property

The mathematical representation of the time shifting property is given by:

$$\text{DTFS}(x[n - M]) = \text{DTFS}(x[n]) \cdot e^{-j\frac{2\pi}{N}kM}$$

Example Illustrating Time Shifting Property

Let's consider a discrete-time signal:

$$x[n] = [1, 2, 3, 4]$$

The DTFS coefficients of x[n] are given by:

$$X[k] = [10, -2j, -2, -2j]$$

Now, let's shift x[n] by 2 samples to obtain a new signal y[n]:

$$y[n] = x[n - 2]$$

The DTFS coefficients of y[n] can be calculated as:

$$Y[k] = X[k] \cdot e^{-j\frac{2\pi}{N}kM}$$

Frequency Shifting Property

The frequency shifting property of the DTFS states that shifting a discrete-time signal in the frequency domain corresponds to multiplying its DTFS coefficients by a complex exponential function.

Explanation of Frequency Shifting Property

The frequency shifting property can be understood as follows: if we have a discrete-time signal x[n] with DTFS coefficients X[k], and we shift x[n] in the frequency domain by L harmonics to obtain a new signal y[n] as:

$$y[n] = e^{j\frac{2\pi}{N}Ln} \cdot x[n]$$

then the DTFS coefficients of y[n] can be expressed as:

$$Y[k] = X[k - L]$$

where N is the period of the signal.

Mathematical Representation of Frequency Shifting Property

The mathematical representation of the frequency shifting property is given by:

$$\text{DTFS}(e^{j\frac{2\pi}{N}Ln} \cdot x[n]) = \text{DTFS}(x[n]) \cdot e^{j\frac{2\pi}{N}Ln}$$

Example Illustrating Frequency Shifting Property

Let's consider a discrete-time signal:

$$x[n] = [1, 2, 3, 4]$$

The DTFS coefficients of x[n] are given by:

$$X[k] = [10, -2j, -2, -2j]$$

Now, let's shift x[n] in the frequency domain by 1 harmonic to obtain a new signal y[n]:

$$y[n] = e^{j\frac{2\pi}{N}n} \cdot x[n]$$

The DTFS coefficients of y[n] can be calculated as:

$$Y[k] = X[k - 1]$$

Time Scaling Property

The time scaling property of the DTFS states that scaling a discrete-time signal in the time domain corresponds to scaling its DTFS coefficients.

Explanation of Time Scaling Property

The time scaling property can be understood as follows: if we have a discrete-time signal x[n] with DTFS coefficients X[k], and we scale x[n] by a factor of L to obtain a new signal y[n] as:

$$y[n] = x[Ln]$$

then the DTFS coefficients of y[n] can be expressed as:

$$Y[k] = X[\frac{k}{L}]$$

where N is the period of the signal.

Mathematical Representation of Time Scaling Property

The mathematical representation of the time scaling property is given by:

$$\text{DTFS}(x[Ln]) = \text{DTFS}(x[n])$$

$$Y[k] = X[\frac{k}{L}]$$

Example Illustrating Time Scaling Property

Let's consider a discrete-time signal:

$$x[n] = [1, 2, 3, 4]$$

The DTFS coefficients of x[n] are given by:

$$X[k] = [10, -2j, -2, -2j]$$

Now, let's scale x[n] by a factor of 2 to obtain a new signal y[n]:

$$y[n] = x[2n]$$

The DTFS coefficients of y[n] can be calculated as:

$$Y[k] = X[\frac{k}{2}]$$

Convolution Property

The convolution property of the DTFS states that the DTFS of the convolution of two signals is equal to the product of their individual DTFS representations.

Explanation of Convolution Property

The convolution property can be understood as follows: if we have two discrete-time signals x1[n] and x2[n] with DTFS coefficients X1[k] and X2[k] respectively, and we convolve x1[n] and x2[n] to obtain a new signal y[n] as:

$$y[n] = x1[n] * x2[n]$$

then the DTFS coefficients of y[n] can be expressed as:

$$Y[k] = X1[k] \cdot X2[k]$$

Mathematical Representation of Convolution Property

The mathematical representation of the convolution property is given by:

$$\text{DTFS}(x1[n] * x2[n]) = \text{DTFS}(x1[n]) \cdot \text{DTFS}(x2[n])$$

Example Illustrating Convolution Property

Let's consider two discrete-time signals:

$$x1[n] = [1, 2, 3, 4]$$ $$x2[n] = [5, 6, 7, 8]$$

The DTFS coefficients of x1[n] and x2[n] are given by:

$$X1[k] = [10, -2j, -2, -2j]$$ $$X2[k] = [26, -6j, -6, -6j]$$

Now, let's convolve x1[n] and x2[n] to obtain a new signal y[n]:

$$y[n] = x1[n] * x2[n]$$

The DTFS coefficients of y[n] can be calculated as:

$$Y[k] = X1[k] \cdot X2[k]$$

Application of Discrete Time Fourier Series

The DTFS finds various applications in Signals & Systems. Let's explore some of its key applications.

Analysis of Periodic Signals

The DTFS can be used to analyze periodic signals by decomposing them into their harmonic components.

Explanation of Analysis of Periodic Signals

The analysis of periodic signals using the DTFS involves the following steps:

1. Obtain the DTFS coefficients of the periodic signal using the DFT algorithm.
2. Identify the dominant harmonic components and their corresponding amplitudes and phases.
3. Determine the fundamental frequency and period of the signal.

Mathematical Representation of Analysis of Periodic Signals

The mathematical representation of the analysis of periodic signals using the DTFS is given by:

$$x[n] = \sum_{k=0}^{N-1} X[k]e^{j\frac{2\pi}{N}kn}$$

where:

• x[n] is the periodic discrete-time signal
• N is the period of the signal
• X[k] is the complex amplitude of the k-th harmonic component
• j is the imaginary unit

Example Illustrating Analysis of Periodic Signals

Let's consider a periodic discrete-time signal:

$$x[n] = [1, 2, 3, 4, 1, 2, 3, 4]$$

The period of the signal is N = 4. The DTFS coefficients of x[n] can be obtained using the DFT algorithm.

Filtering and Signal Processing

The DTFS can be used for filtering and signal processing applications.

Explanation of Filtering and Signal Processing

Filtering and signal processing using the DTFS involve the following steps:

1. Obtain the DTFS coefficients of the input signal using the DFT algorithm.
2. Manipulate the DTFS coefficients to achieve the desired filtering or signal processing operation.
3. Reconstruct the filtered or processed signal using the inverse DTFS.

Mathematical Representation of Filtering and Signal Processing

The mathematical representation of filtering and signal processing using the DTFS is given by:

$$y[n] = \sum_{k=0}^{N-1} Y[k]e^{j\frac{2\pi}{N}kn}$$

where:

• y[n] is the filtered or processed discrete-time signal
• N is the period of the signal
• Y[k] is the complex amplitude of the k-th harmonic component
• j is the imaginary unit

Example Illustrating Filtering and Signal Processing

Let's consider an input discrete-time signal:

$$x[n] = [1, 2, 3, 4]$$

The DTFS coefficients of x[n] are given by:

$$X[k] = [10, -2j, -2, -2j]$$

Now, let's apply a low-pass filter to the input signal by setting the higher frequency components to zero:

$$Y[k] = \begin{cases} X[k], & \text{if } k < K_c \ 0, & \text{otherwise} \end{cases}$$

where K_c is the cutoff frequency. The filtered signal y[n] can be reconstructed using the inverse DTFS.

Spectrum Analysis

The DTFS can be used for spectrum analysis to determine the frequency content of a discrete-time signal.

Explanation of Spectrum Analysis

Spectrum analysis using the DTFS involves the following steps:

1. Obtain the DTFS coefficients of the input signal using the DFT algorithm.
2. Plot the magnitude and phase spectra of the signal to visualize its frequency content.
3. Identify the dominant frequency components and their corresponding amplitudes and phases.

Mathematical Representation of Spectrum Analysis

The mathematical representation of spectrum analysis using the DTFS is given by:

$$X[k] = |X[k]|e^{j\angle X[k]}$$

where:

• X[k] is the complex amplitude of the k-th harmonic component
• |X[k]| is the magnitude of the k-th harmonic component
• \angle X[k] is the phase of the k-th harmonic component

Example Illustrating Spectrum Analysis

Let's consider a discrete-time signal:

$$x[n] = [1, 2, 3, 4]$$

The DTFS coefficients of x[n] are given by:

$$X[k] = [10, -2j, -2, -2j]$$

The magnitude and phase spectra of x[n] can be calculated as:

$$|X[k]| = [10, 2, 2, 2]$$ $$\angle X[k] = [0, -\frac{\pi}{2}, \pi, \frac{\pi}{2}]$$

Advantages and Disadvantages of Discrete Time Fourier Series

The DTFS has several advantages and disadvantages that are important to consider.

Advantages

The advantages of using the DTFS include:

1. Compact representation: The DTFS allows us to represent periodic signals using a finite number of complex amplitudes, which results in a concise representation.
2. Frequency analysis: The DTFS provides insights into the frequency content of a signal, enabling us to analyze and manipulate its spectral characteristics.
3. Signal processing applications: The DTFS is widely used in various signal processing applications such as filtering, modulation, and spectrum analysis.

Disadvantages

The disadvantages of using the DTFS include:

1. Limited to periodic signals: The DTFS is applicable only to periodic signals. It cannot be directly applied to aperiodic or non-periodic signals.
2. Discrete frequency representation: The DTFS represents the frequency content of a signal using discrete frequency components, which may result in a loss of information for signals with continuous frequency spectra.
3. Computational complexity: The computation of the DTFS coefficients using the DFT algorithm can be computationally intensive for large signals or high-resolution frequency analysis.

Conclusion

In conclusion, the discrete time Fourier series (DTFS) is a powerful tool in Signals & Systems for analyzing and manipulating discrete-time signals. It possesses several properties that allow us to manipulate signals effectively, such as the linearity, time shifting, frequency shifting, time scaling, and convolution properties. The DTFS finds applications in various areas, including the analysis of periodic signals, filtering and signal processing, and spectrum analysis. While the DTFS offers advantages such as compact representation and frequency analysis, it also has limitations, including its applicability to periodic signals only and the discrete frequency representation. Understanding the properties and applications of the DTFS is essential for mastering the field of Signals & Systems.

Summary

The discrete time Fourier series (DTFS) is a mathematical tool used in Signals & Systems to analyze and manipulate discrete-time signals. It allows us to represent a periodic discrete-time signal as a sum of complex exponential functions. The DTFS possesses properties such as linearity, time shifting, frequency shifting, time scaling, and convolution. It finds applications in the analysis of periodic signals, filtering and signal processing, and spectrum analysis. The advantages of the DTFS include compact representation, frequency analysis, and signal processing applications. The disadvantages of the DTFS include its limitation to periodic signals, discrete frequency representation, and computational complexity.

Analogy

Imagine you have a collection of musical instruments, each producing a different sound. The discrete time Fourier series (DTFS) is like a magical device that can break down any song played by these instruments into its individual musical notes. By analyzing the notes, you can understand the melody, rhythm, and other characteristics of the song. Similarly, the DTFS allows us to analyze and manipulate discrete-time signals by decomposing them into their harmonic components.