Seeing Signals in Time & Frequency

01 / 10

One signal, three incompatible bandwidth claims

How can three careful engineers report three different widths for the same burst?

Illustrative engineering case: a fictional condition-monitoring node has a 20.0 kbit/s source near an illustrative 2.450 GHz carrier. Its 256-symbol, rectangularly held BPSK-like teaching burst is passed unchanged to a filter designer, a systems engineer, and a compliance engineer. Each can produce a valid number while answering a different question. The failure begins only when the number travels without its definition.

Derived · long-record factor8.859 kHz · 3 dB2 × 0.442946 × Rs from sinc² = 1/2; not a complete filter requirement.
Derived · analytic factor20.000 kHz · null-to-nullFirst zeros at ±10 kHz for the 10 ksymbol/s rectangular pulse factor.
Simulated · tf-lens/1.064.922 kHz · 99%Narrowest corrected-power bin interval in this 256-symbol Hann-windowed record, with ±39.0625 Hz edge uncertainty.

Think about itIf the record doubles but the symbol transitions do not change, which feature narrows?

Answer

The frequency-bin spacing narrows because the observation lasts twice as long. The rectangular-symbol pulse factor keeps its nominal first zeros at ± the symbol rate.

The traces are generated from the pinned rectangular-symbol and deterministic PRBS-9 models, not drawn decoration. The long-record envelope describes the pulse factor. Each finite-record calculation also carries N, fs = 80 ksample/s, periodic Hann window, two-sided corrected power, and its own bin spacing. The 99% value uses this lesson's narrowest-bin rule, not the ITU equal-tail emission method.

Evidence Derived and simulated from P02-S1 Fourier results and the pinned tf-lens/1.0 DFT; ITU-R SM.328-12 is used only to mark the emission-terminology boundary.

Minimum metadata before a bandwidth number

Signal class, observation interval, reference plane, one- or two-sided spectrum convention, units/normalization, and bandwidth definition or method.

02 / 10

Classify the signal and observation first

Are we describing the signal, a statistical process, or one finite look at either one?

Fourier series, Fourier transforms, power spectral density, and the DFT are related tools, not interchangeable labels. The signal class and the observation decide which description is meaningful; the units often reveal a mismatch before the algebra does.

Periodic power signalContinuous 10 kHz toneFourier-series lines or impulses in an ideal spectrum; finite average power, infinite total energy.
Finite-energy signalOne pulseA continuous Fourier transform; if x is volts, X carries volt-seconds before normalization.
Periodic power signalPulse trainFourier-series lines on a repetition-rate grid, shaped by coefficients from one period.
Stationary/random processNoise-like sourcePower spectral density expresses mean power per hertz under stated assumptions.
Finite observationSeeded PRBS recordA deliberately windowed record and calculated DFT, not a claim that the underlying process is finite-energy.

$\begin{aligned}\bar P&=\frac1{T_0}\int_0^{T_0}|x(t)|^2\,\mathrm dt\\E&=\int_{-\infty}^\infty|x(t)|^2\,\mathrm dt\end{aligned}$ T₀ is the period [s]. These are signal power/energy measures: normalized-amplitude² and normalized-amplitude²·s here; voltage requires a stated resistance before calling the result watts or joules.

An ideal eternal tone cannot have both finite duration and a single infinitely narrow frequency. Its ideal transform therefore uses impulses. A real analyzer never observes eternity: it reports a finite, filtered, sampled, and windowed estimate.

Think about itChoose the representation: periodic pulse train, isolated pulse, stationary noise, and one 2048-sample record.

Answer

Use a Fourier series for the ideal periodic train, a Fourier transform for the isolated finite-energy pulse, a PSD for the stationary random process, and a DFT/FFT for the stated finite sample record. The choices can support one another, but their outputs and units are not interchangeable.

Common misconceptionAn FFT is the spectrum of the signal, full stop.

It is the DFT of a particular finite record under a particular window and normalization. It may estimate useful properties of the signal, but those extra claims need conditions.

Evidence Definitions and signal classifications: P02-S1, Chapters 1, 3, and 4; open cross-check: P02-S3.

03 / 10

Complex exponentials as a coordinate system

Why does a real cosine need coefficients at positive and negative frequency?

A complex exponential is a rotating coordinate. Its real-axis projection is a cosine. Pair equal rotations in opposite directions and their imaginary parts cancel, leaving one real waveform—not two emitted carriers and not negative energy.

$e^{j2\pi f_0t}=\cos(2\pi f_0t)+j\sin(2\pi f_0t)$ cos(2πf₀t) = [eʲ²πᶠ⁰ᵗ + e⁻ʲ²πᶠ⁰ᵗ] / 2. Here f₀ is in hertz, t is in seconds, and j² = −1.

0°+f: +1 · −f: +1Half-sum real projection: +1
90°+f: +j · −f: −jHalf-sum real projection: 0
180°+f: −1 · −f: −1Half-sum real projection: −1
270°+f: −j · −f: +jHalf-sum real projection: 0

For any real-valued time signal, the complete statement is conjugate symmetry: $X(-f)=X^*(f)$ . Magnitudes are symmetric, while phases have the matching sign reversal needed to reconstruct a real signal.

Common misconceptionNegative frequency is a second physical oscillator with negative energy.

Negative frequency is part of the complex coordinate description. In a real cosine, the +f and −f coefficients form the conjugate pair required for one real oscillation.

Evidence Definition and symmetry result under the declared transform: P02-S1, Chapters 1 and 4.

04 / 10

Fourier series for repetition

Which part of a repeated pulse sets line spacing, and which part sets the envelope?

A waveform that repeats every T₀ can be built from harmonically related complex exponentials. Repetition sets the allowed line grid. The shape within one period sets the coefficient envelope over that grid.

$\begin{aligned}a_k&=\frac1{T_0}\int_0^{T_0}x(t)e^{-jk\omega_0t}\,\mathrm dt\\\text{Line spacing}&=\frac1{T_0}\end{aligned}$ For periodic x(t), integrate over any complete period T₀ [s]; k is an integer, ω₀ = 2π/T₀ [rad/s], and aₖ has the same units as x. Pulse width controls the sinc-like envelope; T₀ controls line spacing.

Worked micro-example

10% duty pulses at 1 kHz repetition

Under the ideal periodic model, spectral lines are 1 kHz apart. Halving the repetition period doubles that spacing. Narrowing the individual pulse broadens its sinc envelope, but it does not independently set the line spacing.

Think about itIf only the repetition period is halved, what changes?

Answer

Harmonic spacing doubles because 1/T₀ doubles. Holding pulse width fixed leaves the pulse-factor zero spacing fixed, although the sampled line locations on that envelope change.

Think about itIf only the individual pulse width is halved, what changes?

Answer

The pulse-factor zero spacing doubles because it scales as 1/Tpulse. Holding the repetition period fixed leaves the Fourier-series line spacing unchanged.

Evidence Definition and derived 1 kHz line spacing: P02-S1, Chapter 3.

05 / 10

Fourier transform for an isolated waveform

What does one finite event require from its continuous frequency description?

Use the portfolio convention below: frequency is in hertz and the forward transform has no scale factor. If x(t) is volts and time is seconds, then X(f) has volt-seconds before any display normalization.

$\begin{aligned}X(f)&=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}\,\mathrm dt\\x(t)&=\int_{-\infty}^{\infty}X(f)e^{j2\pi ft}\,\mathrm df\end{aligned}$ For a rectangular pulse: X(f) = AT sinc(fT)e⁻ʲ²πᶠᵗ⁰, with sinc(x) = sin(πx)/(πx). A is pulse amplitude [V here], T and t₀ are seconds, and f is hertz.

Checked example

Pulse duration sets the first zeros

A 100 µs rectangular pulse has first magnitude zeros at ±10 kHz. Doubling the duration to 200 µs moves them to ±5 kHz. Longer in time is narrower in frequency; the energy and peak scaling must still be handled separately.

Think about itIf that pulse is delayed without being clipped by the observation, what changes?

Answer

The transform gains the phase factor e⁻ʲ²πᶠᵗ⁰. Its ideal magnitude stays unchanged.

Common misconceptionA shorter event uses less bandwidth because it sends less energy.

Duration and energy are different questions. Shortening the event broadens its transform; changing amplitude or duration also changes energy according to the stated waveform.

Evidence Definition and rectangular-pulse derivation: P02-S1, Chapters 4–5; open cross-check: P02-S3.

06 / 10

Transform properties as prediction tools

Can we predict the spectrum before evaluating another integral?

These properties are an engineering sketchbook. They let you predict direction and structure before calculation, which is often the fastest way to catch an impossible plot.

Think about itIf two time waveforms are added, what happens to their transforms?

Answer

Their transforms add with the same coefficients. Linearity lets a composite waveform be predicted from known parts.

Think about itIf a pulse moves later without clipping, what changes?

Answer

The ideal magnitude stays fixed; phase gains the linear term −2πft₀.

Think about itIf x(t) is compressed to x(2t), what changes?

Answer

The frequency shape expands by 2 and its transform amplitude is multiplied by 1/2.

Think about itIf a waveform is differentiated, which spectral region is emphasized?

Answer

Multiplication by j2πf weights higher |f| more strongly and rotates phase by the sign-dependent factor.

Think about itIf an otherwise continuing waveform is multiplied by a finite gate, what happens?

Answer

Its spectrum is convolved with the gate spectrum, producing finite-observation spreading.

Think about itIf a baseband waveform is multiplied by cos(2πfct), what appears?

Answer

Two half-scaled copies appear around +fc and −fc under the declared transform convention.

Portfolio transform convention · frequency in hertz
Property	Time	Frequency	Prediction
Linearity	a x(t) + b y(t)	a X(f) + b Y(f)	Build complicated signals from known parts.
Time shift	x(t − t₀)	X(f)e⁻ʲ²πᶠᵗ⁰	Change phase, not ideal magnitude.
Time scaling	x(at)	\|a\|⁻¹X(f/a)	Shorter events spread in frequency.
Differentiation	dx(t)/dt	j2πfX(f)	Rapid changes receive greater spectral weight.
Multiplication	x(t)y(t)	X(f) * Y(f)	A gate convolves the underlying spectrum.
Modulation	x(t)cos(2πfct)	[X(f−fc)+X(f+fc)]/2	Translate copies around a carrier.

A finite burst is an underlying waveform multiplied by a gate. Multiplication in time is convolution in frequency, so the gate spreads and reshapes the finite-record result. Carrier multiplication translates scaled copies around ±fc.

Here a and b are scalar coefficients, t₀ is a delay in seconds,a in x(at) is a nonzero dimensionless time-scale factor, and fc is a carrier frequency in hertz. These identities use the transform pair declared in section 5 and assume the transforms exist in the appropriate ordinary or generalized sense.

Go deeperDifferentiation gives orientation, not free gain

Differentiation corresponds to multiplication by j2πf. It emphasizes rapid change and high-frequency content, but a real differentiator also has bandwidth, noise, stability, and implementation limits.

Evidence Definition and transform-property consequences: P02-S1, Chapters 4–5; P02-S3 lectures S16–S18.

07 / 10

A finite record is a windowed observation

Which features belong to the waveform, and which were created by how long we looked?

$x_{o}(t) = x(t)w(t) \Longleftrightarrow X_{o}(f) = X(f) * W(f)$ w is the observation window. For N samples at fs [sample/s], Trecord = N/fs [s] and Δf = fs/N = 1/Trecord [Hz]. Bin spacing is not a universal resolving-power claim.

A rectangular record ends abruptly and therefore has relatively high sidelobes. A periodic Hann record tapers to reduce those sidelobes, at the cost of a wider main lobe and greater noise-equivalent bandwidth. Neither window changes the source waveform; each changes the observation.

On the 2048-point, 80 ksample/s grid, 1.250 kHz is exactly signed bin ±32 because $\frac{1250}{39.0625}=32$ . A 1.270 kHz tone falls at 32.512 bins, so it is off-bin and its finite-record energy spreads according to the selected window response.

10 ksymbol/s rectangular BPSK · 8 samples/symbol · fs = 80 ksample/s
Record	N	Time	Δf	Pulse-factor first zeros
256 symbols	2048	25.6 ms	39.0625 Hz	±10 kHz
512 symbols	4096	51.2 ms	19.53125 Hz	±10 kHz

Reasonableness check

The time support doubles while transition time stays fixed. Fine frequency sampling should improve, but the broad pulse-shaped envelope should not contract by 2×.

Common misconceptionA longer FFT narrows the signal.

A longer record narrows Δf. It can reveal finer structure or separate some nearby tones, but it does not automatically narrow the underlying waveform's occupied support.

Evidence Definition and derivation: P02-S1 discrete Fourier material and P02-S3; numeric fixtures independently checked from fs/N.

08 / 10

Energy, power, spectrum, and PSD normalization

Why can the height of an FFT bin change when the physical tone does not?

Scaling by record length, sample rate, coherent gain, window power, one- or two-sided conversion, and logarithmic units all change plotted numbers. A trustworthy result states those choices and checks a conserved quantity before applying display corrections.

$\int |x(t)|^{2}d t = \int |X(f)|^{2}d f$ DFT form used below: Σ|x[n]w[n]|² = (1/N)Σ|X[k]|² before display correction. The continuous identity applies to finite-energy signals under the declared transform pair.

Do not mix these quantities without a conversion
Quantity	What it emphasizes	Typical condition
Amplitude spectrum	Sinusoidal component amplitude and phase	State DFT/transform scaling and window coherent gain
Energy spectral density	Energy distribution of a finite-energy waveform	State transform convention and physical units
PSD / periodogram	Mean power per hertz	State estimator, averaging, window power, and sidedness
Normalized dB display	Relative shape and dynamic range	0 dB reference and display floor are explicit

The exact discrete normalization used by tf-lens/1.0

The two-sided coefficient amplitude is |X[k]|/(N·CG). Corrected power per bin is |X[k]|²/(NΣw²), in normalized-amplitude² per bin, and the displayed PSD column divides that result by Δf to give normalized-amplitude²/Hz. The plot then divides corrected bin power by its largest bin; 0 dB is that local maximum and −120 dB is only a display floor.

Common misconceptionA taller bin after increasing N proves that the tone gained power.

Not without a consistent normalization. The same tone can concentrate into a different number of bins or be scaled by a different DFT convention while its physical power remains unchanged.

A normalization mistake does not stay cosmetic: it changes integrated noise, contaminates calculated occupied bandwidth, and can propagate into SNR, EVM, and later instrument comparisons.

Evidence Parseval and spectral-quantity definitions: P02-S1, Chapters 4–7; implementation formulas are derived above and regression tested.

09 / 10

Bandwidth is a defined measurement

Which width answers the decision in front of us?

A width is useful only when its rule matches the decision. A mask, a channel allocation, a filter passband, a noise bandwidth, and an occupied-bandwidth result are not synonyms. They may constrain the same design from different directions.

Definition first; number second
Method	Boundary rule	Valid use here
x dB	Outer limits beyond which spectrum is at least x dB below a declared 0 dB reference	Normative terminology; the reference and spectrum quantity must be named
Null-to-null	Distance between identifiable analytic or stably bracketed zeros	Derived rectangular-pulse factor only when the zeros are meaningful
99% calculated finite record	Narrowest contiguous bins containing at least 99% of corrected bin power	Local deterministic comparison with ±one-bin edge uncertainty
Necessary bandwidth	System width sufficient for required information rate and quality under stated conditions	Terminology only; this lesson does not calculate it
Channel / mask	Compare the defined waveform result with an applicable allocation or limiting curve	Deferred until the jurisdiction, standard, plane, detector, and method are known
Noise-equivalent bandwidth	Ideal rectangular bandwidth passing the same white-noise power as the actual response	Noise integration and window/filter correction, not waveform occupancy

Normative terminology boundary. ITU-R SM.328-12 defines occupied bandwidth with equal out-of-band mean-power tails. The lens intentionally implements the prompt-pinned narrowest contiguous 99% bin interval instead. It therefore reports a calculated finite-record comparison, never an ITU occupied-bandwidth measurement or compliance result.

Owned later. Alias folding belongs to 02.4, root-raised-cosine design to 02.6, analyzer operation to 08.3, and regulatory limits to Paths 07/09. This module supplies the representation and measurement vocabulary those later decisions require.

Choose the method from the engineering question
Question	Use	Do not silently substitute
Will the wanted signal pass?	Amplitude and phase requirements across wanted support	One 3 dB number
How much mean power lies in a span?	A cited occupied-bandwidth method	Analytic first nulls
How much white noise passes?	Noise-equivalent bandwidth	Occupied or 3 dB width
What sampling plan is credible?	Occupied support plus analog transition band and guard	Nominal symbol rate alone

Interactive · tf-lens/1.0 · deterministic finite-record model

Time/Frequency Lens

Change one part of the waveform or observation, predict the consequence, and use a named bandwidth only when it answers the engineering question.

If the PRBS record doubles while rate and pulse shape stay fixed, what narrows?

Fine bin spacingPulse envelopeBothNeither

Choose a prediction before changing the record length.

Then switch the observation window and compare the solid and dashed spectra before choosing a stakeholder metric.

Bin spacing Δf: 39.0625 Hz
Record / transform: N = 2048
Coherent gain: 0.5000
Window ENBW: 1.500 bins
99% calculated occupied width: 64.922 kHz

Time record · ideal waveform-source plane D2 · normalized amplitude

PRBS-9 BPSK burst, 256 rate intervals, 8 samples/interval.

Underlying discrete model
Windowed observation

Two-sided normalized power spectrum

Current hann result overlaid with rectangular; −120 dB display floor.

hann
rectangular

Active cause: The default result combines a 256-symbol PRBS-9 BPSK observation with a periodic Hann window.

Observation-window trade
Window	Coherent gain	ENBW	Coherent-tone main lobe	Peak sidelobe
Rectangular	1.000	1.000 bins	2 bins, null-to-null	about −13.3 dB
Periodic Hann	0.500	1.500 bins	4 bins, null-to-null	about −31.5 dB

Bin -64: -2.500 kHz, -0.29 dB normalized power
Bin -50: -1.953 kHz, -0.00 dB normalized power
Bin -5: -195.313 Hz, -1.97 dB normalized power
Bin -2: -78.125 Hz, -0.75 dB normalized power
Bin -1: -39.063 Hz, -0.84 dB normalized power
Bin 0: 0.000e+0 Hz, -1.51 dB normalized power
Bin 1: 39.063 Hz, -0.84 dB normalized power
Bin 2: 78.125 Hz, -0.75 dB normalized power

Engineering question

Decision guidance: A cited occupied-bandwidth method fits the occupancy question. This model's 99% result is calculated, not a formal instrument or regulatory result. Your choice: Fit: the 99% calculated finite-record method answers this model-comparison question when its record, plane, and uncertainty travel with the value.

Static decision fallback · available without JavaScript
Stakeholder question	Required method	Why the other widths fail
Wanted-signal filter	Amplitude and phase limits across wanted support	A single 3 dB, null, or occupied width omits response shape
Spectral occupancy comparison	Cited 99% calculated method, record, plane, and uncertainty	3 dB and null-to-null use different boundaries
White-noise integration	Noise-equivalent bandwidth	Waveform occupancy does not preserve passed noise power
Sampling plan	Occupied support plus analog transition and guard	Symbol rate or one width alone omits the analog boundary

Inspect the accessible numeric tables

33 representative time samples
n	Time	Raw	Windowed
0	0.000e+0 s	1.0000	0.0000
64	800.000 µs	1.0000	0.0096
128	1.600 ms	1.0000	0.0381
192	2.400 ms	-1.0000	-0.0843
256	3.200 ms	-1.0000	-0.1464
320	4.000 ms	-1.0000	-0.2222
384	4.800 ms	-1.0000	-0.3087
448	5.600 ms	1.0000	0.4025
512	6.400 ms	1.0000	0.5000
576	7.200 ms	1.0000	0.5975
640	8.000 ms	1.0000	0.6913
704	8.800 ms	1.0000	0.7778
768	9.600 ms	1.0000	0.8536
832	10.400 ms	1.0000	0.9157
896	11.200 ms	1.0000	0.9619
960	12.000 ms	1.0000	0.9904
1024	12.800 ms	-1.0000	-1.0000
1087	13.588 ms	-1.0000	-0.9907
1151	14.387 ms	1.0000	0.9625
1215	15.188 ms	-1.0000	-0.9166
1279	15.988 ms	-1.0000	-0.8546
1343	16.788 ms	1.0000	0.7791
1407	17.587 ms	-1.0000	-0.6928
1471	18.388 ms	-1.0000	-0.5990
1535	19.188 ms	-1.0000	-0.5015
1599	19.987 ms	-1.0000	-0.4040
1663	20.788 ms	1.0000	0.3101
1727	21.587 ms	1.0000	0.2235
1791	22.388 ms	-1.0000	-0.1475
1855	23.188 ms	-1.0000	-0.0851
1919	23.987 ms	1.0000	0.0386
1983	24.788 ms	1.0000	0.0099
2047	25.587 ms	1.0000	0.0000

Strongest calculated spectral bins
Signed bin	Frequency	Two-sided amplitude	Power / bin	PSD / Hz	Relative dB
-64	-2.500 kHz	0.142859	0.01360572	3.4831e-4	-0.292
-50	-1.953 kHz	0.147737	0.01455088	3.7250e-4	-0.000
-5	-195.313 Hz	0.117739	0.00924170	2.3659e-4	-1.971
-2	-78.125 Hz	0.135555	0.01225002	3.1360e-4	-0.748
-1	-39.063 Hz	0.134046	0.01197883	3.0666e-4	-0.845
0	0.000e+0 Hz	0.124119	0.01027037	2.6292e-4	-1.513
1	39.063 Hz	0.134046	0.01197883	3.0666e-4	-0.845
2	78.125 Hz	0.135555	0.01225002	3.1360e-4	-0.748
5	195.313 Hz	0.117739	0.00924170	2.3659e-4	-1.971
30	1.172 kHz	0.117028	0.00913038	2.3374e-4	-2.024
50	1.953 kHz	0.147737	0.01455088	3.7250e-4	0.000
64	2.500 kHz	0.142859	0.01360572	3.4831e-4	-0.292

Download deterministic CSV

Export rounding is pinned: sample rate 6 decimals, bin frequency 9, amplitude/power 12, PSD 15, relative dB 6, and time 12.

X[k] = Σ x[n]w[n]e^−j2πkn/N · Δf = fs/N · ENBW = NΣw²/(Σw)²Two-sided coefficient amplitude = |X[k]|/(N · coherent gain). Corrected power/bin = |X[k]|²/(NΣw²); PSD = power/bin ÷ Δf. Relative display power is normalized to the strongest bin and floored at −120 dB. Parseval relative residual before display correction: < 1.000e-12.

Boundary: normalized deterministic calculation at the ideal waveform-source plane D2; source information originates at D0. Samples are dimensionless, not volts or watts. Samples/interval selects a native discrete computational grid—no analog sampling operation is modeled, so alias folding is outside this lesson by construction. PRBS-9 repeats after 511 bits, so a 512-symbol request begins a second sequence period. This is not a regulatory emission result, not a sampled-hardware prediction, and not an instrument simulation. Local narrowest-bin interval from -36.855 kHz to 28.066 kHz; this is not the ITU equal-tail emission measurement.

JavaScript is unavailable, so the controls are inactive. The server-rendered default remains: PRBS-9 BPSK, 10 ksymbol/s, 256 symbols, 8 samples/symbol, N = 2048, fs = 80 ksample/s, Δf = 39.0625 Hz, periodic Hann coherent gain 0.5, ENBW 1.5 bins, and a two-sided 99% calculated finite-record width of 64.922 kHz ± one bin. The visible tables above remain the text alternative without hydration.

Bin spacing 39.0625 hertz. ENBW 1.500 bins. 99% calculated occupied width 64.922 kHz.

Common misconceptionThe Hann window removes leakage for free.

It trades lower sidelobes for a wider main lobe and 1.5-bin ENBW under this periodic convention, compared with 1.0 bin for a rectangular observation.

Evidence Normative terminology: ITU-R SM.328-12 (09/2025), RR-derived §§1.152–1.153 and Recommendation §1.8; transform/window behavior: P02-S1. The local 99% algorithm is explicitly non-normative.

10 / 10

Update the waveform decision record

What can we now state without pretending that the final modulation has already been chosen?

The recurring condition-monitoring case starts with a 20.0 kbit/s information stream and an illustrative 2.450 GHz carrier. This first record deliberately stops before choosing a final mapping or pulse shape.

Information-source class · D0: Finite framed 20.0 kbit/s payload bursts; source statistics and exact framing remain unresolved.
Burst / repetition assumptions: Teaching record: seeded PRBS-9, 256 symbols at 10 ksymbol/s; actual burst cadence and idle behavior remain open.
Observation · D2: 25.6 ms ideal waveform-source record, N = 2048 at 80 ksample/s, before RF filtering or hardware.
Spectrum convention: Two-sided signed frequency in hertz; X[k] uses the stated negative-exponent DFT and fftshift order.
Units / normalization: Dimensionless normalized samples; periodic Hann; CG = 0.5; ENBW = 1.5 bins; corrected power and display reference stated.
Bandwidth question / method: Filter response, local 99% calculated occupancy, noise ENBW, sampling guard, and later mask comparison remain distinct fields.
Checked precursor / uncertainty: Rectangular 10 ksymbol/s BPSK-like factor: ±10 kHz first nulls, 20 kHz total; finite-record edges carry ±Δf.
Still unknown: Final mapping, I/Q convention, pulse shaping, sample plan, impairment limits, applicable standard, and formal measurement method.

Think about itOne waveform, three stakeholders: which method belongs in each handoff, and which two tempting substitutes must be rejected?

Answer

Filter designer: provide amplitude and phase requirements across wanted support. Reject a bare 3 dB width and the local 99% span because neither specifies passband ripple, phase, or rejection.
Systems occupancy comparison: use the local 99% calculated finite-record method with N, fs, window, normalization, plane, and ±Δf. Reject analytic null-to-null and 3 dB widths because they answer different boundary questions.
Compliance engineer: use the applicable standard/channel mask and its measurement method at the required RF plane. Reject this lesson's local 99% calculation and analytic null-to-null width as compliance evidence.

Evidence Illustrative condition-monitoring case from the approved portfolio baseline; rectangular-symbol nulls are derived from P02-S1; regulatory method and limits remain deferred.

Handoff to 02.2

We can now predict how information occupies time and frequency. Next we ask why and where that information should be translated or mapped onto an RF carrier at all.

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01What changes in the magnitude spectrum when an isolated pulse is delayed?
Model answer
The ideal transform magnitude does not change; the transform acquires a linear phase factor. That answer assumes an equivalent observation window and no clipping at the record boundary.
02A record contains 2048 samples at 80 ksample/s. What is its frequency-bin spacing?
Model answer
Δf = fs/N = 80,000/2048 = 39.0625 Hz.
03Why can rectangular- and Hann-window FFTs of the same tone look different?
Model answer
The windows multiply different finite observations, so their spectral responses redistribute the same tone differently. The underlying tone did not change.
04Which bandwidth belongs in a white-noise integration through a filter?
Model answer
Noise-equivalent bandwidth (ENBW). A 3 dB width or occupied-bandwidth result alone does not preserve integrated white-noise power.
05An off-bin tone raises many FFT bins. Is that necessarily new transmitter noise?
Model answer
No. First test coherent sampling, another record length, or another window. Finite-record leakage can redistribute a deterministic tone across bins.
06What must accompany a reported bandwidth number?
Model answer
The signal and observation class, reference plane, spectrum/PSD convention and normalization, bandwidth definition or method, and any unresolved assumptions.

Sources and further study

The lesson cites and paraphrases these sources. P02-S4 was rechecked on 5 September 2026: SM.328-12 (09/2025) remains in force, while SM.328-11 is superseded. No plot on this page is measured evidence.

Primary technical references

P02-S1 · Informative: A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed., Chapters 3–7.
P02-S2 · Informative: J. G. Proakis and M. Salehi, Digital Communications, 5th ed., Chapters 2, 4, and 5.
P02-S3 · Informative: MIT OpenCourseWare: Unified Engineering Signals and Systems, especially S15–S18 and S21–S22.
P02-S4 · Normative terminology: ITU-R SM.328-12: Spectra and bandwidth of emissions, approved 2025-09-01 and in force when checked.

Derived and simulated evidence

Definition: transform/DFT sign, two-sided convention, normalized sample units, and window corrections are printed beside use.
Derived: pulse zeros, 3 dB pulse-factor width, bin spacing, coherent gain, ENBW, and Parseval identities follow the shown equations.
Simulated: tf-lens/1.0 uses the pinned PRBS-9 sequence and deterministic FFT; fixed-table rounding is regression tested.
Illustrative: the condition-monitoring node is fictional and names no wireless standard.

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