Fourier Series, Spectral Density, Bandwidth, and DTFT Symmetry

The Discrete Fourier Series

A periodic sequence $x[n]$ with period $N$ can be represented exactly by $N$ harmonically related complex exponentials:

\[\tilde X[k]=\sum_{n=0}^{N-1}\tilde x[n]e^{-j2\pi kn/N},\qquad k=0,1,\dots,N-1\] \[\tilde x[n]=\frac{1}{N}\sum_{k=0}^{N-1}\tilde X[k]e^{j2\pi kn/N},\qquad n=0,1,\dots,N-1\]

These are the discrete Fourier series (DFS) pair. They are identical in form to the DFT — the only difference is interpretation: the DFS describes a periodic sequence, the DFT describes one period of a finite sequence.

The basis set ${e^{j2\pi kn/N}}$ has exactly $N$ distinct members, because

\[e^{j2\pi(k+N)n/N}=e^{j2\pi kn/N}e^{j2\pi n}=e^{j2\pi kn/N}\]

So the representation is finite and exact. This is fundamentally different from the continuous-time Fourier series, which needs infinitely many harmonics.

From Fourier Series to the Fourier Transform

Domain	Periodic	Aperiodic
Continuous time	Fourier series (discrete spectrum)	Fourier transform (continuous spectrum)
Discrete time	DFS/DFT (discrete, periodic spectrum)	DTFT (continuous, periodic spectrum)

Making a signal periodic in one domain makes its spectrum discrete in the other. This duality is the whole reason the DFT exists.

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Power Density Spectrum

For a power signal, total energy is infinite, so we cannot talk about an energy spectrum. Instead we use average power and its distribution over frequency.

Parseval for Power Signals

\[P=\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\vert x[n]\vert^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_{xx}(e^{j\omega})\,d\omega\]

where $S_{xx}(e^{j\omega})$ is the power density spectrum (also written $P_{xx}(\omega)$).

For a periodic signal with period $N$:

\[P=\frac{1}{N}\sum_{n=0}^{N-1}\vert x[n]\vert^2=\sum_{k=0}^{N-1}\vert c_k\vert^2\]

where $c_k=\tilde X[k]/N$ are the Fourier coefficients. This is Parseval’s theorem for the DFS: average power in time equals the sum of squared coefficient magnitudes.

Example. $x[n]=\cos(2\pi n/8)$.

\[P=\frac{1}{2}\]

The two Fourier coefficients at $k=1$ and $k=7$ each have magnitude $\tfrac12$, and $\left(\tfrac12\right)^2+\left(\tfrac12\right)^2=\tfrac12$ ✓.

The Wiener–Khinchine Theorem

The power density spectrum is the DTFT of the autocorrelation sequence:

\[S_{xx}(e^{j\omega})=\sum_{l=-\infty}^{\infty}r_{xx}[l]\,e^{-j\omega l}\] \[r_{xx}[l]=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_{xx}(e^{j\omega})e^{j\omega l}\,d\omega\]

This is why autocorrelation and spectrum are two views of the same information. Every spectrum estimator in practice is an autocorrelation estimator in disguise.

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Energy Density Spectrum

For a finite-energy signal:

\[E=\sum_{n=-\infty}^{\infty}\vert x[n]\vert^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}\vert X(e^{j\omega})\vert^2\,d\omega\]

The quantity

\[S_{xx}(e^{j\omega})=\vert X(e^{j\omega})\vert^2\]

is the energy density spectrum. It is non-negative, real, and even for a real signal.

Signal type	Total measure	Density function	Units
Energy signal	$E$ (joules)	$\vert X(e^{j\omega})\vert^2$	energy/rad
Power signal	$P$ (watts)	$S_{xx}(e^{j\omega})$	power/rad
Periodic	$P$ from Fourier coefficients	line spectrum at harmonics	power per line

Exam distinction: an energy signal has $P=0$ and a continuous energy density; a power signal has $E=\infty$ and a power density. Never mix the two integrals.

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Bandwidth

There is no single definition of bandwidth. The definition depends on the application.

1. Absolute (Null-to-Null) Bandwidth

The width between the first zeros of the spectrum. For a rectangular pulse of length $N$, the main lobe of the sinc spans

\[B_{\text{null}}=\frac{2}{N}\quad\text{(in radians/sample)}\]

2. 3-dB Bandwidth

The width where the power is half the peak value (magnitude $1/\sqrt2$). Standard for filters.

3. Percentage Bandwidth

For a signal whose spectrum is concentrated in $F_1\le F\le F_2$, the $95\%$ bandwidth is the width containing 95% of the energy or power:

\[B_{95}=F_2-F_1 \quad\text{such that}\quad \int_{F_1}^{F_2}S(f)\,df=0.95\,E_{\text{total}}\]

You also see 75%, 90%, and 99% versions. This is the definition used for real measured signals where the spectrum has no clean edges.

4. Equivalent Noise Bandwidth

The width of an ideal rectangular filter with the same peak gain and the same integrated noise power:

\[B_{\text{eq}}=\frac{1}{\vert H_{\max}\vert^2}\int_0^{\infty}\vert H(f)\vert^2\,df\]

5. Bandlimited Signals

A signal is bandlimited if its spectrum is zero outside $\vert F\vert\le B$. That is the condition under which the sampling theorem applies and exact reconstruction is possible.

Time–Bandwidth Product

The shorter a pulse in time, the wider its bandwidth:

\[\Delta t\cdot\Delta f\ge\frac{1}{4\pi}\quad\text{(Gaussian)}\]

For a rectangular pulse of duration $T$: $B_{\text{null}}=2/T$. This is the transform’s uncertainty principle, and it is why you cannot have arbitrarily sharp time resolution and arbitrary frequency resolution at the same time.

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Frequency Ranges of Natural Signals

Some context for choosing a sampling rate (from Proakis Table 4.2):

Signal	Typical frequency range
Speech	50 Hz – 4 kHz (telephone: 300–3400 Hz)
Music / audio	20 Hz – 20 kHz
ECG	0.05 – 150 Hz
EEG	0.1 – 100 Hz
EMG	10 Hz – 1 kHz
Sonar	10 Hz – 1 MHz
Radar	1 MHz – 100 GHz
Visible light	$4\times10^{14}$ – $7.5\times10^{14}$ Hz

The Nyquist rate follows directly: audio needs $\ge40$ kHz (hence 44.1 kHz CD), telephony needs $\ge8$ kHz (hence 8 kHz PCM), ECG needs only a few hundred Hz.

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DTFT Symmetry Properties

For a sequence decomposed into even and odd parts, $x[n]=x_e[n]+x_o[n]$:

Time domain	Frequency domain
$x[n]$ real	$X(e^{j\omega})=X^*(e^{-j\omega})$ (conjugate symmetric)
$x[n]$ real and even	$X(e^{j\omega})$ real and even
$x[n]$ real and odd	$X(e^{j\omega})$ purely imaginary and odd
$x[n]$ purely imaginary	$X(e^{j\omega})=-X^*(e^{-j\omega})$ (conjugate antisymmetric)
$x_e[n]$	$\operatorname{Re}{X(e^{j\omega})}$
$x_o[n]$	$j\operatorname{Im}{X(e^{j\omega})}$

The Four Symmetries in Full

For a real $x[n]$, write $X(e^{j\omega})=X_R(\omega)+jX_I(\omega)$:

\(X_R(\omega)=X_R(-\omega) \quad\text{(even)}\) \(X_I(\omega)=-X_I(-\omega) \quad\text{(odd)}\) \(\vert X(e^{j\omega})\vert=\vert X(e^{-j\omega})\vert \quad\text{(even magnitude)}\) \(\angle X(e^{j\omega})=-\angle X(e^{-j\omega}) \quad\text{(odd phase)}\)

Practical consequences:

1. A real signal’s spectrum is fully described by $\omega\in[0,\pi]$; the rest is redundant.
2. The DFT of a real $N$-point sequence satisfies $X[N-k]=X^*[k]$, so only $N/2+1$ values are independent.
3. This is exactly why real-input FFT algorithms can halve the work — they pack two real sequences into one complex FFT and unpack the conjugate symmetry afterwards.

Worked Check

$x[n]=\delta[n]+\delta[n-1]$ is real and not even and not odd. Its transform:

\[X(e^{j\omega})=1+e^{-j\omega}=e^{-j\omega/2}\left(e^{j\omega/2}+e^{-j\omega/2}\right)=2\cos(\omega/2)e^{-j\omega/2}\]

Magnitude $2\cos(\omega/2)$ is even ✓. Phase $-\omega/2$ is odd ✓. Consistent with conjugate symmetry.

$x[n]=\delta[n+1]+\delta[n-1]$ is real and even:

\[X(e^{j\omega})=e^{j\omega}+e^{-j\omega}=2\cos\omega\]

Real and even ✓.

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Summary

1. The DFS is the DFT applied to a periodic sequence; $N$ harmonics represent it exactly.
2. Power signals use a power density spectrum; energy signals use $\vert X\vert^2$.
3. Wiener–Khinchine: the power spectrum is the DTFT of the autocorrelation.
4. Bandwidth has at least five definitions — always state which one.
5. Time and bandwidth are inversely related; you cannot beat the uncertainty principle.
6. Real signals have conjugate-symmetric spectra, which halves the independent data.