Signals and Systems

3.7 Properties of Discrete-Time Fourier Series

Imron Rosyadi

Properties of Discrete-Time Fourier Series

Section 3.7

Today, we delve into the properties of the Discrete-Time Fourier Series, or DTFS. Understanding these properties is crucial for analyzing and manipulating discrete-time signals efficiently. Many of these properties will seem familiar, as they have strong parallels with the Continuous-Time Fourier Series properties we’ve already covered. However, we’ll also highlight some key differences that are unique to the discrete-time domain. These properties not only offer conceptual insights but also simplify the computation of Fourier series coefficients for complex signals.

Overview of DTFS Properties

DTFS properties share strong similarities with CTFS properties.

They are essential tools for signal analysis and manipulation.

Note

Shorthand Notation:

We use the notation \(x[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} a_k\) to indicate that \(x[n]\) is a periodic signal with period \(N\) and \(a_k\) are its Fourier series coefficients.

The table on the next slide summarizes the most important properties. It’s important to remember this shorthand notation, as it’s commonly used in textbooks and research to express the Fourier series relationship.

Table 3.2: Properties of Discrete-Time Fourier Series

Property	Periodic Signal	Fourier Series Coefficients
Periodicity	\(x[n]\) periodic with period \(N\)	\(a_k\) periodic with period \(N\)
Linearity	\(A x[n]+B y[n]\)	\(A a_{k}+B b_{k}\)
Time Shifting	\(x\left[n-n_{0}\right]\)	\(a_{k} e^{-j k(2 \pi / N) n_{0}}\)
Frequency Shifting	\(e^{j M(2 \pi / N) n} x[n]\)	\(a_{k-M}\)
Conjugation	\(x^{*}[n]\)	\(a_{-k}^{*}\)
Time Reversal	\(x[-n]\)	\(a_{-k}\)
Periodic Convolution	\(\sum_{r=\langle N\rangle} x[r] y[n-r]\)	\(N a_{k} b_{k}\)
Multiplication	\(x[n] y[n]\)	\(\sum_{l=\langle N\rangle} a_{l} b_{k-l}\)
First Difference	\(x[n]-x[n-1]\)	\(\left(1-e^{-j k(2 \pi / N)}\right) a_{k}\)
Running Sum	\(\sum_{m=-\infty}^{n} x[m]\) (\(a_0=0\) for periodicity)	\(\frac{1}{\left(1-e^{-j k(2 \pi / N)}\right)} a_{k}\) (for \(k \neq 0, \pm N, \ldots\))
Conjugate Symmetry (Real \(x[n]\))	\(x[n]\) real	\(a_{k}=a_{-k}^{*}\), \(|a_k|=|a_{-k}|\), \(\angle a_k = -\angle a_{-k}\)
Real & Even Signals	\(x[n]\) real and even	\(a_k\) real and even
Real & Odd Signals	\(x[n]\) real and odd	\(a_k\) purely imaginary and odd

This table summarizes the key properties. I’ve slightly reordered some for logical flow, but the content is the same as in the textbook. We’ll discuss some of the most important properties in more detail, especially those with distinct discrete-time characteristics, and walk through examples.

Time Shifting Property

If \(x[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} a_k\), then \(x[n-n_0] \stackrel{\mathcal{FS}}{\longleftrightarrow} a_k e^{-j k(2\pi/N)n_0}\).

A time shift in the signal domain corresponds to a phase shift in the frequency domain.

Let’s visualize this with a simple square wave.

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

viewof P_shift = Inputs.number([1, 15], {
  label: "P (Pulse Width, 2N1+1):",
  step: 1,
  value: 3
})

viewof n0_shift = Inputs.range([0, N_shift - 1], {
  label: "Shift (n0):",
  step: 1,
  value: 0
})

Observe the top plot: as you adjust the n0 slider, the original square wave x[n] shifts to x[n-n0]. Now look at the bottom plot for the Fourier coefficients. The magnitudes of the coefficients do not change with a time shift. This makes sense, as shifting a signal doesn’t change its energy distribution across frequencies. However, the phases of the coefficients change linearly with k. This linear phase shift is directly proportional to the amount of time shift n0. This property is fundamental for understanding how delays affect the frequency content of signals.

Multiplication Property

If \(x[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} a_k\) and \(y[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} b_k\), both periodic with period \(N\).

Then \(x[n] y[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} d_k\), where \(d_k\) is given by periodic convolution:

\[ d_k = \sum_{l=\langle N\rangle} a_l b_{k-l} \]

Important

Key Difference from CTFS:

In CTFS, multiplication in time domain corresponds to aperiodic convolution of coefficients.

In DTFS, it corresponds to periodic convolution of coefficients, meaning the summation is over \(N\) terms and coefficients \(b_k\) are considered periodic.

The multiplication property is one of the most important distinctions between CTFS and DTFS. While both involve convolution in the frequency domain, DTFS uses periodic convolution. This means that when we calculate \(b_{k-l}\), we treat \(b\) as a periodic sequence. This property is crucial in understanding how mixing signals in the time domain affects their spectral components, such as in modulation.

First Difference Property

If \(x[n] \stackrel{\mathcal{FS}}{\longleftrightarrow} a_k\), then \(x[n]-x[n-1] \stackrel{\mathcal{FS}}{\longleftrightarrow} (1-e^{-j k(2\pi/N)}) a_k\).

This is the discrete-time parallel to the differentiation property in continuous time.

Tip

This property is useful when the first difference of a signal is simpler to analyze than the original signal itself, often simplifying coefficient calculation.

Let’s see how the first difference affects the spectrum of a discrete-time ramp signal.

// Define N for DT ramp
viewof N_diff = Inputs.number([1, 20], {
  label: "N (Period):",
  step: 1,
  value: 10
})

In the top plot, you see a periodic ramp signal and its first difference. The first difference of a ramp is a constant (except for the wrap-around point), which is a much simpler signal. Observe the bottom plot: the magnitudes of the coefficients for the first difference signal are significantly altered compared to the original ramp. This property effectively scales the coefficients by a frequency-dependent term, often leading to a “flatter” or simpler spectrum for signals with sharp transitions when differenced. For example, the DC component (\(k=0\)) of the difference signal is very small (ideally zero for a perfect step), which is reflected in its coefficient.

Parseval’s Relation for DT Periodic Signals

\[ \frac{1}{N} \sum_{n=\langle N\rangle}|x[n]|^{2}=\sum_{k=\langle N\rangle}\left|a_{k}\right|^{2} \]

This relation states that the average power in one period of \(x[n]\) (LHS) equals the sum of the average powers in all its harmonic components (RHS).

Tip

This is a powerful conservation-of-energy principle, linking the time-domain energy (or power) of a signal to its frequency-domain representation. It applies to both CT and DT signals, with appropriate changes in summation/integration and scaling factors.

Let’s verify Parseval’s relation for a discrete-time square wave.

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

viewof P_parseval = Inputs.number([1, 15], {
  label: "P (Pulse Width, 2N1+1):",
  step: 1,
  value: 3
})

The output clearly shows that the calculated LHS and RHS are equal, verifying Parseval’s relation for this discrete-time square wave. This confirms that the total average power of the signal is conserved whether you measure it in the time domain or sum the powers of its frequency components. This is a fundamental concept in signal processing for power analysis.

Example 3.13: Linearity

Find the Fourier series coefficients \(a_k\) of \(x[n]\) shown below.

\(x[n]\) can be decomposed into \(x_1[n]\) (square wave) and \(x_2[n]\) (DC component).

Tip

Strategy:

1. Decompose \(x[n]\) into simpler components whose coefficients are known or easy to find.
2. Use the linearity property to sum the coefficients.

graph TD
    A["x[n]"] --> B{Linearity Property}
    B --> C["x1[n]"]
    B --> D["x2[n]"]
    C --> E["b_k"]
    D --> F["c_k (DC component)"]
    E & F --> G["a_k = b_k + c_k"]

This example demonstrates the power of the linearity property. Instead of directly calculating the Fourier series for the complex signal \(x[n]\), we can break it down into two simpler signals: a standard square wave \(x_1[n]\) and a DC offset \(x_2[n]\). We already know how to find the coefficients for these simpler signals from previous examples or basic definitions. Then, we just add their coefficients to get the coefficients for \(x[n]\).

Example 3.13: Visualizing Components and Coefficients

\(x[n]\) with \(N=5\). \(x_1[n]\) is a square wave with \(N_1=1\), and \(x_2[n]\) is a DC component.

The top plot shows the original signal \(x[n]\) (orange), its square wave component \(x_1[n]\) (blue), and its DC component \(x_2[n]\) (green). You can clearly see how \(x[n]\) is the sum of \(x_1[n]\) and \(x_2[n]\). The bottom plot shows the magnitudes of their respective Fourier coefficients. Notice how \(|a_k|\) (orange) is the sum of \(|b_k|\) (blue) and \(|c_k|\) (green) for each \(k\). Specifically, for \(k=0\), \(a_0 = b_0 + c_0 = 3/5 + 1 = 8/5\). For other \(k\), \(c_k=0\), so \(a_k = b_k\). This visual confirms the linearity property in action.

Example 3.14: Minimum Power & Signal Reconstruction

Problem: Determine \(x[n]\) given:

1. \(x[n]\) is periodic with \(N=6\).
2. \(\sum_{n=0}^{5} x[n]=2\).
3. \(\sum_{n=2}^{7}(-1)^{n} x[n]=1\).
4. \(x[n]\) has the minimum power per period among signals satisfying 1-3.

Solution Steps:

• From Fact 2: \(a_0 = \frac{1}{N} \sum x[n] = \frac{1}{6}(2) = \frac{1}{3}\).
• From Fact 3: \((-1)^n = e^{-j\pi n} = e^{-j(2\pi/6)3n}\), so this term corresponds to \(k=3\). Thus, \(a_3 = \frac{1}{N} \sum (-1)^n x[n] = \frac{1}{6}(1) = \frac{1}{6}\).
• From Fact 4 (Minimum Power): Parseval’s relation states \(P = \sum_{k=\langle N\rangle} |a_k|^2\). To minimize power, all other coefficients (\(a_1, a_2, a_4, a_5\)) must be zero.

Therefore, \(x[n]\) only has \(a_0\) and \(a_3\) as non-zero coefficients.

\(x[n] = a_0 e^{j0(2\pi/6)n} + a_3 e^{j3(2\pi/6)n} = a_0 + a_3 e^{j\pi n} = \frac{1}{3} + \frac{1}{6}(-1)^n\).

This example is a classic problem that ties together several properties. We use the definition of the Fourier series coefficients to extract \(a_0\) and \(a_3\) from the given summations. Then, the critical insight comes from Parseval’s relation: to minimize the total average power, any coefficient not explicitly determined must be zero. This allows us to uniquely determine the signal \(x[n]\) from partial information about its spectrum. Let’s visualize this reconstructed signal.

Example 3.14: Visualizing Reconstructed \(x[n]\)

The reconstructed signal: \(x[n] = \frac{1}{3} + \frac{1}{6}(-1)^n\).

Here’s the signal \(x[n]\) that satisfies all the given conditions. It’s a simple, alternating sequence, which makes sense given that only the DC component (\(a_0\)) and the highest frequency component (\(a_3\) for \(N=6\)) were non-zero. This demonstrates how spectral information, combined with power minimization, can uniquely define a signal.

Example 3.15: Periodic Convolution

Problem: Determine and sketch \(w[n]\) given its Fourier series coefficients \(c_k\). \(w[n]\) is periodic with \(N=7\), and \(c_k = \frac{\sin^2(3\pi k/7)}{7\sin^2(\pi k/7)}\).

Strategy:

1. Recognize \(c_k\) as a product of known coefficient forms.
2. Use the periodic convolution property to find \(w[n]\).

We observe \(c_k = 7 d_k^2\), where \(d_k\) are the coefficients of a square wave \(x[n]\) with \(N=7\) and \(N_1=1\) (pulse width \(P=3\)).

Since \(c_k = 7 d_k d_k\), by the periodic convolution property (\(N a_k b_k \longleftrightarrow \sum x[r]y[n-r]\)), we have:

\[ w[n] = \sum_{r=\langle 7\rangle} x[r] x[n-r] \]

This means \(w[n]\) is the periodic convolution of the square wave \(x[n]\) with itself.

This example takes a different approach: starting from the Fourier coefficients and working back to the time-domain signal. The key is recognizing the structure of \(c_k\). It looks like the square of the coefficients of a standard discrete-time square wave. This immediately points us to the periodic convolution property. So, \(w[n]\) is the periodic convolution of a square wave with itself. This is a powerful technique for finding signals whose spectra are products of known spectra. Next, we’ll visualize the square wave and its self-convolution.

Example 3.15: Visualizing Periodic Convolution

\(x[n]\) is a square wave with \(N=7\) and \(P=3\) (\(N_1=1\)). \(w[n]\) is its periodic convolution with itself: \(w[n] = (x * x)[n]\).

The top plot shows the original square wave \(x[n]\) with \(N=7\) and a pulse width of 3. The bottom plot shows the result of its periodic convolution with itself, \(w[n]\). Notice how the convolution operation “smoothes” the signal, creating a triangular shape (a discrete-time triangle wave). This is a common result when convolving a pulse with itself, and seeing it derived through the frequency domain (by squaring the coefficients) is a powerful illustration of the convolution property.

Summary of DTFS Properties

• Periodicity: Both \(x[n]\) and \(a_k\) are periodic with period \(N\).
• Linearity: \(A x[n]+B y[n] \longleftrightarrow A a_{k}+B b_{k}\).
• Time Shift: \(x[n-n_0] \longleftrightarrow a_k e^{-j k(2\pi/N)n_0}\) (phase shift).
• Multiplication: \(x[n] y[n] \longleftrightarrow \sum_{l=\langle N\rangle} a_l b_{k-l}\) (periodic convolution of coefficients).
• Periodic Convolution: \(\sum_{r=\langle N\rangle} x[r] y[n-r] \longleftrightarrow N a_k b_k\) (product of coefficients).
• First Difference: \(x[n]-x[n-1] \longleftrightarrow (1-e^{-j k(2\pi/N)}) a_k\).
• Parseval’s Relation: Average power is conserved across domains. \[ \frac{1}{N} \sum_{n=\langle N\rangle}|x[n]|^{2}=\sum_{k=\langle N\rangle}\left|a_{k}\right|^{2} \]

Caution

Always remember the periodic nature of DTFS coefficients and the finite summation in periodic convolution, as these are key distinctions from CTFS.

In summary, the properties of the Discrete-Time Fourier Series provide a powerful toolbox for signal analysis. They allow us to understand how operations in the time domain affect the frequency domain, and vice-versa. The finite nature of the series and the periodicity of its coefficients are central to all these properties. Mastering these properties will be essential as we move on to more complex signal processing concepts. Thank you.