Signals and Systems

CT vs. DT Fourier Series: A Comparative Recap

Imron Rosyadi

CT vs. DT Fourier Series: A Comparative Recap

Understanding the Core Differences

Welcome back! Today, we’ll consolidate our understanding of Fourier series by directly comparing its continuous-time and discrete-time versions. While they share a common goal of decomposing periodic signals into a sum of harmonically related exponentials, their mathematical forms and inherent properties differ significantly. This comparison will highlight the unique characteristics of each and reinforce why these distinctions are important in ECE.

Continuous-Time Fourier Series (CTFS) Recap

For a periodic CT signal \(x(t)\) with fundamental period \(T_0\) and fundamental frequency \(\omega_0 = 2\pi/T_0\).

Synthesis Equation:

(Reconstruction from coefficients)

\[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_0 t} \]

Analysis Equation:

(Calculation of coefficients)

\[ a_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \omega_0 t} dt \]

Note

Key Characteristics:

• Infinite series: Requires summing an infinite number of terms.
• Convergence issues: Partial sums may not perfectly reconstruct \(x(t)\), especially at discontinuities.
• Gibbs phenomenon: Overshoots/undershoots near discontinuities in partial sums.

Let’s quickly review the CTFS. The synthesis equation shows that any periodic continuous-time signal can be expressed as an infinite sum of complex exponentials. The analysis equation provides the formula to calculate the coefficients \(a_k\) by integrating the signal over one period, weighted by a complex exponential. The key takeaways here are the infinite nature of the series, which leads to potential convergence issues and the infamous Gibbs phenomenon at discontinuities when only a finite number of terms are used for reconstruction.

Discrete-Time Fourier Series (DTFS) Recap

For a periodic DT signal \(x[n]\) with fundamental period \(N\) and fundamental frequency \(\omega_0 = 2\pi/N\).

Synthesis Equation:

(Reconstruction from coefficients)

\[ x[n] = \sum_{k=\langle N\rangle} a_k e^{j k \omega_0 n} = \sum_{k=\langle N\rangle} a_k e^{j k(2\pi/N)n} \]

Analysis Equation:

(Calculation of coefficients)

\[ a_k = \frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-j k \omega_0 n} = \frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-j k(2\pi/N)n} \]

Note

Key Characteristics:

• Finite series: Sums only \(N\) distinct terms.
• No convergence issues: The finite sum perfectly reconstructs \(x[n]\).
• No Gibbs phenomenon: Exact reconstruction with all \(N\) terms.
• Coefficients are periodic: \(a_k = a_{k+N}\).

Now, for the DTFS. Again, we have synthesis and analysis equations, but with crucial differences. The summation in the synthesis equation is explicitly shown as a finite sum over \(N\) consecutive values of \(k\). The analysis equation involves a summation over \(N\) samples of the signal, rather than an integral. The most important characteristics are the finite nature of the series, which means no convergence issues and no Gibbs phenomenon. Also, remember that the coefficients \(a_k\) themselves are periodic with period \(N\).

CTFS vs. DTFS: A Side-by-Side Comparison

Continuous-Time Fourier Series (CTFS)

• Signal: \(x(t)\) (continuous-time)
• Period: \(T_0\)
• Fundamental Freq: \(\omega_0 = 2\pi/T_0\)
• Series Type: Infinite sum \[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j k \omega_0 t} \]
• Coefficient Calculation: Integral \[ a_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \omega_0 t} dt \]
• Convergence: Requires conditions, partial sums approximate \(x(t)\).
• Gibbs Phenomenon: Present at discontinuities for partial sums.
• Coefficients periodicity: \(a_k\) are generally not periodic.

Discrete-Time Fourier Series (DTFS)

• Signal: \(x[n]\) (discrete-time)
• Period: \(N\)
• Fundamental Freq: \(\omega_0 = 2\pi/N\)
• Series Type: Finite sum (N terms) \[ x[n] = \sum_{k=\langle N\rangle} a_k e^{j k(2\pi/N)n} \]
• Coefficient Calculation: Summation \[ a_k = \frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-j k(2\pi/N)n} \]
• Convergence: Always exact with \(N\) terms.
• Gibbs Phenomenon: Not present.
• Coefficients periodicity: \(a_k\) are periodic with period \(N\).

This slide provides a concise summary of the key differences. Notice the fundamental distinction between integrals and summations, and the implications of infinite versus finite series. The periodicity of coefficients in DTFS is also a unique and important property.

Example: Periodic Square Wave (CTFS)

Consider a CT periodic square wave \(x(t)\) with period \(T_0\), amplitude \(A=1\), and pulse width \(T_1\).

Let \(T_0=1\) and \(T_1=0.5\).

The Fourier series coefficients are given by:

\[ a_k = \frac{A T_1}{T_0} \text{sinc}\left(k \frac{T_1}{T_0}\right) = \frac{A T_1}{T_0} \frac{\sin(\pi k T_1/T_0)}{\pi k T_1/T_0} \]

For \(k=0\), \(a_0 = \frac{A T_1}{T_0}\).

To illustrate these concepts, let’s use a common signal: the periodic square wave. For continuous time, the coefficients are given by a sinc function. This means the spectrum is continuous-like, decaying as \(k\) increases, with zero crossings. Let’s visualize this.

CTFS: Square Wave Visualization

Here we visualize the CT square wave and its Fourier coefficients.

viewof A_ct = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})

viewof T0_ct = Inputs.number([0.1, 5], {
  label: "T0 (Fundamental Period):",
  step: 0.1,
  value: 1
})

viewof T1_ct = Inputs.number([0.1, 2], {
  label: "T1 (Pulse Width):",
  step: 0.1,
  value: 0.5
})

The top plot shows the continuous-time periodic square wave. The bottom plot shows the magnitudes of its Fourier series coefficients. Notice that the coefficients form a continuous envelope (the sinc function), but are only defined at discrete harmonic frequencies \(k\omega_0\). This spectrum extends to infinity, with the magnitudes decreasing as \(k\) increases.

Example: Periodic Square Wave (DTFS)

Consider a DT periodic square wave \(x[n]\) with period \(N\), amplitude \(A=1\), and pulse width \(P\).

Let \(N=10\) and \(P=5\) (i.e., \(2N_1+1=5 \implies N_1=2\)).

The Fourier series coefficients are given by:

\[ a_k = \frac{A}{N} \frac{\sin(\pi k P/N)}{\sin(\pi k/N)}, \quad k \neq 0, \pm N, \pm 2N, \ldots \]

For \(k=0, \pm N, \ldots\):

\[ a_k = \frac{A P}{N} \]

Now, let’s look at the discrete-time equivalent. We define a square wave in discrete time, with period \(N\) and pulse width \(P\). The formula for its coefficients is a ratio of sines, which is the discrete-time equivalent of the sinc function. Let’s visualize this.

DTFS: Square Wave Visualization

Here we visualize the DT square wave and its Fourier coefficients.

viewof A_dt = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})

viewof N_dt = Inputs.number([1, 20], {
  label: "N (Period):",
  step: 1,
  value: 10
})

viewof P_dt = Inputs.number([1, 15], {
  label: "P (Pulse Width, 2N1+1):",
  step: 1,
  value: 5
})

Here, we see the discrete-time square wave and its Fourier coefficients. Notice that the signal \(x[n]\) is a sequence of points. The coefficients \(|a_k|\) are also a sequence of points, and there are only \(N\) distinct coefficients (from \(k=0\) to \(N-1\)). This spectrum is periodic, meaning the pattern of coefficients from \(0\) to \(N-1\) repeats for \(N\) to \(2N-1\), and so on. This is a direct consequence of the discrete nature of the time signal.

Visual Comparison: CTFS vs. DTFS Coefficients

CTFS Coefficients (k continuous envelope)

viewof A_ct_comp = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})

viewof T0_ct_comp = Inputs.number([0.1, 5], {
  label: "T0 (Fundamental Period):",
  step: 0.1,
  value: 1
})

viewof T1_ct_comp = Inputs.number([0.1, 2], {
  label: "T1 (Pulse Width):",
  step: 0.1,
  value: 0.5
})

DTFS Coefficients (k sampled points)

viewof A_dt_comp = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})

viewof N_dt_comp = Inputs.number([1, 20], {
  label: "N (Period):",
  step: 1,
  value: 10
})

viewof P_dt_comp = Inputs.number([1, 15], {
  label: "P (Pulse Width, 2N1+1):",
  step: 1,
  value: 5
})

Here’s a direct visual comparison of the coefficient magnitudes. The CTFS coefficients follow a continuous sinc envelope, meaning that if we were to take more and more harmonics, they would fill out this shape. The DTFS coefficients, however, are discrete samples of a similar envelope, but they are also periodic. This “sampling” of the frequency domain is a direct consequence of the time-domain signal being discrete.

Reconstruction & Gibbs Phenomenon (CTFS)

Using only a finite number of terms to reconstruct a CT square wave will show the Gibbs phenomenon.

viewof A_ct_rec = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})
viewof T0_ct_rec = Inputs.number([0.1, 5], {
  label: "T0 (Fundamental Period):",
  step: 0.1,
  value: 1
})
viewof T1_ct_rec = Inputs.number([0.1, 2], {
  label: "T1 (Pulse Width):",
  step: 0.1,
  value: 0.5
})
viewof K_rec_ct = Inputs.range([1, 20], {
  label: "Number of Harmonics (K):",
  step: 1,
  value: 1
})

This interactive plot demonstrates the reconstruction of a CT square wave. Adjust the K slider to change the number of harmonics used in the reconstruction. As you increase K, the reconstructed signal gets closer to the original, but notice the prominent overshoots and undershoots near the discontinuities. This is the Gibbs phenomenon, which persists even with a large number of terms, though it becomes narrower. It highlights the difficulty of representing sharp transitions with a finite sum of smooth sinusoids.

Reconstruction & No Gibbs Phenomenon (DTFS)

Using all \(N\) terms perfectly reconstructs a DT periodic square wave.

viewof A_dt_rec = Inputs.number([0, 5], {
  label: "A (Amplitude):",
  step: 0.1,
  value: 1
})

viewof N_dt_rec = Inputs.number([1, 20], {
  label: "N (Period):",
  step: 1,
  value: 10
})
viewof P_dt_rec = Inputs.number([1, 15], {
  label: "P (Pulse Width, 2N1+1):",
  step: 1,
  value: 5
})

viewof M_rec_dt = Inputs.range([0, 4], {
  label: "Number of Terms (M):",
  step: 1,
  value: 0
})

In contrast to CTFS, the DTFS reconstruction behaves very differently. Adjust the M slider, which controls the number of terms used (from 1 to \(N\)). As you increase the number of terms, the reconstructed signal quickly converges. Once you include all \(N\) distinct terms (e.g., when the slider is at its maximum value for \(N=10\)), the reconstructed signal perfectly matches the original signal. There are no Gibbs phenomena or approximation errors. This exact reconstruction is a hallmark of the discrete-time Fourier series.

Conclusion and Key Takeaways

CTFS: The Infinite Spectroscope

• Infinite number of harmonic components.
• Continuous spectrum envelope (sinc for square wave).
• Approximation when using finite terms.
• Gibbs phenomenon at discontinuities.
• Used for analyzing analog signals in continuous systems.

DTFS: The Finite Spectrum Analyzer

• Finite number (\(N\)) of distinct harmonic components.
• Discrete spectrum (sampled sinc for square wave), which is periodic.
• Exact reconstruction with all \(N\) terms.
• No Gibbs phenomenon.
• Used for analyzing sampled signals and discrete systems.

Tip

Analogy:

Think of CTFS as trying to describe a smooth painting with an infinite palette of colors, while DTFS is like describing a pixelated image with a finite, repeating set of colors.

To wrap up, remember these core differences. CTFS deals with continuous signals and infinite spectra, leading to approximations and Gibbs phenomenon with finite terms. DTFS deals with discrete signals, has a finite, periodic spectrum, and provides exact reconstruction with all its terms. These distinctions are fundamental to understanding signal processing in both analog and digital domains. Thank you for your attention!