1.7: DTFS Properties

Learning Objectives: Discrete-Time Fourier Series (DTFS) Properties

1. Solve for Fourier series coefficients for new signals using DTFS properties by relating the new signal to signals with known FS coefficients.

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Discrete-Time Fourier Series Properties

DT Fourier Series Properties Examples

Discrete Time Fourier Series Example

Summary

Core DTFS equations

If x[n] is periodic with period N and fundamental frequency equal to \(\omega_0=\dfrac{2\pi}{N}\), we write:

\(x[n] \;\Longleftrightarrow\; a_k\)

where \(\Longleftrightarrow\) represents a Fourier Series.

Synthesis (reconstruction) Equation:

\[ x[n] = \sum_{k=\langle N\rangle} a_k\,e^{j k\omega_0 n} \]

Analysis (coefficients) Equation:

\[ a_k = \frac{1}{N}\sum_{n=\langle N\rangle} x[n]\,e^{-j k\omega_0 n} \]

Key DTFS properties

• Multiplication in time

If \(x[n]\Longleftrightarrow a_k\) and \(y[n]\Longleftrightarrow b_k\) (same period \(N\)), then

\[ x[n]\,y[n] \;\Longleftrightarrow\; d_k = \sum_{\ell=\langle N\rangle} a_\ell\,b_{k-\ell} \]

(This looks like a discrete convolution-style summation in the coefficient index.)

• First difference

If \(x[n]\Longleftrightarrow a_k\), then

\[ x[n]-x[n-1] \;\Longleftrightarrow\; \Bigl(1-e^{-j\frac{2\pi}{N}k}\Bigr)\,a_k \]

• Parseval’s relation (average power)

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

Left side = average power in one period of \(x[n]\). Right side = sum of powers in each harmonic (via \(a_k\)).

Table 1.7.1 Properties of the Discrete-Time Fourier Series (DTFS)

Periodic Signal	Fourier Series Coefficients
\(x[n]\), \(y[n]\) periodic with period \(N\) and fundamental frequency \(\omega_0=\frac{2\pi}{N}\)	\(a_k\), \(b_k\) periodic with period \(N\)
Parseval’s Relation for Periodic Signals
\(\displaystyle \frac{1}{N}\sum_{n=(N)}|x[n]|^2\) = \(\displaystyle \sum_{k=(N)}|a_k|^2\)
Linearity
\(\displaystyle A\,x[n] + B\,y[n]\)	\(\displaystyle A\,a_k + B\,b_k\)
Time Shifting
\(\displaystyle x[n-n_0]\)	\(\displaystyle a_k\,e^{-j\frac{2\pi}{N}kn_0}\)
Frequency Shifting
\(\displaystyle e^{j\frac{2\pi}{N}mn}\,x[n]\)	\(\displaystyle a_{k-m}\)
Conjugation
\(\displaystyle x^*[n]\)	\(\displaystyle a_{-k}^*\)
Time Reversal
\(\displaystyle x[-n]\)	\(\displaystyle a_{-k}\)
Time Scaling
\(\displaystyle x_m[n]= \begin{cases} x[n/m], & \text{if } n \text{ is a multiple of } m\\[2pt] 0, & \text{if } n \text{ is not a multiple of } m \end{cases} \) \(\displaystyle \text{(periodic with period } mN)\)	\(\displaystyle \frac{1}{m}a_k\) \(\displaystyle \text{(viewed as periodic with period } mN)\)
Periodic Convolution
\(\displaystyle \sum_{r=(N)} x[r]\,y[n-r]\)	\(\displaystyle N\,a_k\,b_k\)
Multiplication
\(\displaystyle x[n]\,y[n]\)	\(\displaystyle \sum_{\ell=(N)} a_\ell\,b_{k-\ell}\)
Conjugate Symmetry for Real Signals
\(\displaystyle x[n]\ \text{real}\)	\(\displaystyle a_k=a_{-k}^*\) \(\displaystyle \Re\{a_k\}=\Re\{a_{-k}\}\) \(\displaystyle \Im\{a_k\}=-\Im\{a_{-k}\}\) \(\displaystyle |a_k|=|a_{-k}|\) \(\displaystyle \angle a_k=-\angle a_{-k}\)
Real and Even Signals
\(\displaystyle x[n]\ \text{real and even}\)	\(\displaystyle a_k\ \text{real and even}\)
Real and Odd Signals
\(\displaystyle x[n]\ \text{real and odd}\)	\(\displaystyle a_k\ \text{purely imaginary and odd}\)