Pulse Shaping, ISI & Decisions

01 / 10

Rectangular symbols pass bits but fail the spectrum decision

If every ideal sample is correct, why can the waveform still be a poor transmitter proposal?

Start with a zero-order-held symbol stream. Sampling at the center of each interval recovers the entered values perfectly in this noise-free calculation, yet the discontinuities create a slowly decaying sinc pulse factor. A receiver decision test can pass while a filter or adjacent-use constraint fails.

The sample decisions are perfect because this first calculation adds no noise or channel. The separate illustrative filter proxy asks for no more than −35 dBr beyond ±1.25Rₛ; the rectangular pulse factor is about −14.9 dB there, so it fails that invented design proxy by roughly 20 dB. This is a derived teaching comparison, not an emission mask or measurement.

Think about itWhat does smoothing the transitions gain, and what must it spend?

Answer

It can reduce sidelobes and concentrate more energy near the wanted band. It spends time support, FIR taps, group delay, burst transients, and often more sensitivity to timing and peak behavior.

This connects 02.1's multiplication/convolution view to the isolated decision points owned by planned 02.5. A practical waveform decision must satisfy both views at named planes.

Evidence Rectangular pulse transform and finite-gating context: P02-S1; PAM waveform consequences: P02-S2. The −35 dB boundary is explicitly illustrative.

02 / 10

From a symbol sequence to a waveform

What turns isolated mapper outputs into one continuous transmit trajectory?

Place a copy of pulse p(t) at every symbol time and scale it by the corresponding real or complex symbol aₖ. Their sum is the waveform.

$s(t) = \sum _{k} a_{k} p(t - k T_{s})$ aₖ is the D2 symbol, p(t) is the declared pulse, Tₛ = 1/Rₛ [s], k is integer, and s(t) is observed at D3 after the selected TX pulse filter. The model uses normalized complex samples, not volts.

Derived from an ideal α = 0.35 combined raised-cosine pulse so the construction is visible. The four patterned terms are not four channels: they are four shifted contributions to one waveform.

Hold the four symbols fixed; change only p(t)
Pulse	Time behavior	Frequency consequence	Decision caution
Rectangular	One UI, abrupt edges	Sinc factor with slow sidelobe decay	Simple center samples do not prove acceptable spectral behavior
Ideal sinc-like Nyquist	Infinite tails	Minimum ideal support at α = 0	Zero at other integer samples, but unrealizable infinite duration
Finite RRC	Truncated, symmetric FIR	Approaches the selected root-RC magnitude	One RRC alone is not the combined zero-ISI response

Common misconceptionThe constellation points define the transmitted spectrum.

They define desired symbol coordinates at D2. Spectrum also depends on pulse shape, symbol timing, finite burst gating, sequence statistics, and the observation/normalization.

Evidence Definition and construction: Proakis and Salehi, Digital Communications, signal-space/PAM and band-limited-channel treatment; transform context: P02-S1.

03 / 10

Rectangular pulse and sinc cost

Which spectral features belong to one symbol pulse, and which belong to the finite burst observation?

$p(t)=\operatorname{rect}\left(\frac t{T_s}\right)\quad\Longleftrightarrow\quad P(f)=T_s\operatorname{sinc}(fT_s)$ sinc(x) = sin(πx)/(πx). With Rₛ = 1/Tₛ, the first pulse-factor zeros are at f = ±Rₛ and the total null-to-null width is 2Rₛ.

At 10 ksymbol/s, the rectangular pulse factor has first zeros at ±10 kHz and a 20 kHz total null-to-null main lobe. That says nothing universal about occupied bandwidth: its sidelobes continue outside the first zeros. A finite burst then multiplies the continuing symbol waveform by another time gate, convolving this factor with the gate spectrum and adding finer structure.

Think about itIf the burst doubles but Tₛ and the rectangular pulse are unchanged, what moves?

Answer

The finite-record detail becomes finer because the gate lasts longer. The analytic rectangular pulse zeros remain at ±Rₛ; symbol timing did not change.

Common misconceptionSymbol rate equals occupied bandwidth.

Symbol rate sets a time scale. A bandwidth number still needs the pulse, spectrum quantity, reference plane, observation, and boundary method. Even the 2Rₛ rectangular null-to-null result is not an occupied-bandwidth definition.

Terminology boundary. ITU-R SM.328-12 is cited for emission and bandwidth terms. This lesson does not turn analytic pulse support or a finite calculation into a necessary- or occupied-bandwidth result.

Evidence Rectangular transform/gating: P02-S1 and MIT OCW Signals and Systems; terminology boundary: ITU-R SM.328-12 (09/2025).

04 / 10

The zero-ISI condition applies at sample instants

Must neighboring pulses disappear everywhere, or only where the receiver decides?

Let g(t) include the TX filter, channel, and RX filter. At the correct decision timing, one desired pulse must have a declared nonzero center and every shifted neighbor must contribute zero at that instant.

$g(0) = C \ne 0 \text{and} g(n T_{s}) = 0 \text{for every integer} n \ne 0$ This first Nyquist zero-ISI condition applies to the combined response and declared timing. The workbench normalizes the no-channel finite combined center to C = 1 for impulse reporting.

Analytic ideal response. A finite RRC pair only approximates these integer zeros; the checked implementation residual appears in section 7.

Between integer instants, the responses overlap and add. That overlap is how a band-limited pulse spans time. The useful cancellation is coordinated at the samples; it is not an absence of energy between them.

Common misconceptionZero ISI means the pulses never overlap.

Zero ISI means the unwanted shifted responses are zero at the selected decision samples under the stated combined response. They generally overlap everywhere else.

Go deeperThe frequency-domain criterion is a complementary-sum condition

Shifted spectral replicas of the combined pulse must sum to a constant over the decision band. This lesson uses that result only to orient the RC family; proof depth and generalized partial-response signaling are deferred.

Evidence First Nyquist criterion and PAM detection: P02-S2; interpretation cross-check: Sklar, Chapters 1–4.

05 / 10

Raised-cosine roll-off trades bandwidth and time localization

How much ideal spectral support should we spend to shorten the troublesome time tail?

Analytic ideal responses redrawn from the equations. α = 0 minimizes ideal support but has the longest sinc-like time tail; α = 1 doubles the minimum support and localizes the impulse more strongly.

$B_{\mathrm{ideal,NN}} = (1 + \alpha)R_{s} \qquad f_{\mathrm{edge}} = \frac{\pm (1 + \alpha)R_{s}}{2}$ 0 ≤ α ≤ 1. B is the total ideal raised-cosine null-to-null support on a signed complex-baseband axis; it is not automatically occupied, necessary, channel, or measured bandwidth.

Rₛ = 10 ksymbol/s · analytic ideal response
α	Signed edge	Total ideal support	Time-domain implication
0.00	±5.00 kHz	10.00 kHz	Minimum ideal support; sinc-like tail is longest
0.35	±6.75 kHz	13.50 kHz	Canonical first-pass compromise
1.00	±10.00 kHz	20.00 kHz	Widest support; strongest time localization of these cases

Checked node example

α = 0.35 at 10 ksymbol/s

(1 + 0.35) × 10 ksymbol/s = 13.5 kHz total, so each signed complex-baseband edge is ±6.75 kHz. The unit check is hertz because roll-off is dimensionless.

Common misconceptionLower roll-off is free bandwidth savings.

Lower α narrows ideal support but lengthens the impulse tail. A finite implementation then needs more span to control truncation residual and generally leaves less timing margin.

Evidence RC response and Nyquist trade: Proakis/Salehi; bandwidth arithmetic: internal derivation and regression test.

06 / 10

Split the response with root-raised-cosine filters

Why place a square root of the target response on both sides of the channel?

D2Gray QPSK symbolsE[|aₖ|²] = 1

D3Pulse-shaped samples80 ksample/s baseline

R3RX matched-filter decisionscombined-response plane

$|H_{\mathrm{TX,RRC}}(f)|\,|H_{\mathrm{RX,RRC}}(f)|=|H_{\mathrm{RC}}(f)|$ The statement assumes the intended matched split and no intervening distortion beyond the declared channel model. Each finite RRC tap set in pulse-eye/1.0 is normalized to unit discrete energy.

The TX RRC limits the emitted baseband shape; the equal RX filter correlates the received waveform with that pulse. For additive white Gaussian noise and a known pulse/timing, the matched filter maximizes sample-time output SNR. The no-channel TX/RX product approaches the raised-cosine target at R3.

Common misconceptionOne RRC output is already a zero-ISI raised-cosine waveform.

A single RRC has the square-root magnitude response. The zero-ISI target belongs to the combined response. Nor can the matched pair invert an arbitrary echo channel.

Think about itWhere should raw symbol decisions, eye samples, and EVM be reported in this lesson?

Answer

At R3 after the stated RX matched filter and known implementation-delay removal. D3 is appropriate for the transmit sample stream and finite spectrum, not for the final symbol decision.

Evidence Matched-filter AWGN result and RRC split: P02-S2. Plane identifiers and unit-energy convention: Portfolio Conventions v1.0.

07 / 10

Finite filters create delay and residual error

What changes when an infinite pulse family becomes a finite FIR in a burst radio?

$h(x) = \frac{[\sin (\pi (1-\alpha)x) + 4\alpha x \cos (\pi (1+\alpha)x)]}{[\pi x(1-(4\alpha x)^{2})]}$ x = t/Tₛ. The implementation evaluates analytic limits at x = 0 and x = ±1/(4α), then normalizes the finite tap vector so Σ|h[n]|² = 1.

α = 0.35 · span 8 symbols · 8 samples/symbol · 10 ksymbol/s
Quantity	Calculation	Checked result
Taps per RRC	span × sps + 1	65
Delay per RRC	(N − 1)/2	32 samples = 4 UI = 0.4 ms
TX + RX delay	2 × 32 samples	64 samples = 8 UI = 0.8 ms
Tap energy	Σ\|h[n]\|²	1.000000000000
Combined center	g[0] after center normalization	1.000000000000
Nonzero sampled ISI energy	Σₙ≠₀ \|g[nTₛ]\|², n = −8…+8	2.540729794e-4 (-35.950 dB)

Finite deterministic combined-response samples · symmetric neighbor subset
Offset	g(nTₛ), center = 1	Interpretation
-4 UI	1.0491102e-2	Finite truncation residual, not AWGN or channel ISI
-3 UI	-2.6227000e-3	Finite truncation residual, not AWGN or channel ISI
-2 UI	7.1676954e-4	Finite truncation residual, not AWGN or channel ISI
-1 UI	1.2566610e-4	Finite truncation residual, not AWGN or channel ISI
+0 UI	1.0000000e+0	Declared desired center
+1 UI	1.2566610e-4	Finite truncation residual, not AWGN or channel ISI
+2 UI	7.1676954e-4	Finite truncation residual, not AWGN or channel ISI
+3 UI	-2.6227000e-3	Finite truncation residual, not AWGN or channel ISI
+4 UI	1.0491102e-2	Finite truncation residual, not AWGN or channel ISI

A symmetric odd-length FIR delays every retained feature by half its order. At burst edges, the filters do not yet contain a complete symbol history. Metrics must either include those transients deliberately or exclude them with a stated guard. The workbench excludes the first and last span + channelGuard symbols.

Go deeperThe singular values are limits, not exceptions to the pulse equation

At x = 0, h = 1 + α(4/π − 1). At x = ±1/(4α), the finite limit is α/√2 times [(1 + 2/π)sin(π/(4α)) + (1 − 2/π)cos(π/(4α))]. Evaluating these limits avoids 0/0 and keeps all supported α/sps combinations finite.

Evidence RRC equation and delay interpretation: P02-S2; all displayed finite values independently reproduced from the equation and pinned in regression tests.

08 / 10

The eye diagram folds decision history

What does one compact plot reveal about many symbol histories and one timing reference?

One unit interval (UI) is one symbol interval Tₛ. An eye overlays short matched-filter waveform segments on a shared clock phase. Different traces are different local histories; the marked vertical line is the declared decision instant, not an automatically recovered clock.

Vertical opening proxy: 5th-percentile positive component minus 95th-percentile negative component at the selected phase, divided by the ideal level separation.
Horizontal opening proxy: Contiguous interval around nominal phase where that robust component separation remains positive.
Crossing distribution: Linear-interpolated zero crossings of sign-changing I/Q component transitions; mean, standard deviation, and count are shown.
Population / scaling: 256 seeded symbols; first/last guarded edges excluded; unit-average symbol energy; I-eye shown and Q included in numeric proxies for QPSK.

The default calculation uses 8 native samples/UI, nominal phase 0 UI, 30 dB decision-reference SNR, no echo, and R3 after matched filtering. Display interpolation makes the overlay readable; it does not create new simulated samples or a timing-recovery claim.

Common misconceptionAn eye is merely an oscilloscope trigger trick.

It is a projection conditioned on timing reference, filtering, sample density, amplitude scaling, symbol history, and finite population. A measured eye adds instrument, trigger, calibration, and acquisition conditions; this lesson shows deterministic simulation only.

Evidence Eye/Nyquist interpretation: P02-S2 and Sklar. R&S Understanding EVM v01.00 is used only to orient companion constellation/EVM views, not to define this simulated eye metric.

09 / 10

Timing, bandwidth limit, and echoes close the eye differently

If two cases have similar RMS EVM, which view tells you whether more SNR will help?

$c(t) = \delta (t) + a e^{j\theta}\delta (t - \tau)$ a is echo magnitude, θ is relative phase, and τ is delay. The workbench applies a finite DC-normalized Blackman-windowed sinc fractional-delay branch, then adds complex AWGN before the RX RRC.

Same scalar EVM can require different remedies
Cause	Impulse / eye signature	Constellation signature	First remedy category
Short span	Symmetric nonzero neighbor taps; deterministic closure	History-dependent clusters around each ideal point	Longer filter or relaxed roll-off
Timing offset	Marker moves away from best opening; samples leave intended zeros	History-dependent displacement correlated with slope	Timing acquisition/tracking evidence
Complex echo	Delayed/rotated secondary response; asymmetric or multi-path closure	Discrete history-conditioned subclusters	Channel/equalization decision
AWGN	Trace and crossing spread without a deterministic new impulse	Approximately circular local broadening	Link/noise margin

Think about itBefore changing a control, which views should move for +0.20 UI timing, and which should not?

Answer

The decision marker, sampled constellation, EVM, and decisions should move. The D3 spectrum, filter taps, and physical combined impulse remain unchanged because only the sampling phase changed.

Think about itBefore adding a 0.30-amplitude echo at 0.50 UI, what new evidence should appear?

Answer

A delayed secondary impulse component and history-dependent eye/constellation structure. Noise reduction may tighten each cluster but cannot remove the deterministic delayed contribution.

Interactive · pulse-eye/1.0 · deterministic baseband simulation

Pulse & Eye Workbench

Choose the pulse implementation, then separate truncation, timing, echo, and AWGN signatures at the R3 matched-filter decision plane. No recovery or equalizer is hiding the active cause.

Predict before reveal

Which signature should dominate?

Neighbor taps rise symmetricallyThe sample marker moves off the eye centerHistory-dependent skew or a second impulse appearsTraces broaden without a new deterministic echo

First remedy category

Guided cases

Choose a signature and remedy category before revealing a guided case.

Mapping

Symbol rate (ksymbol/s)10 ksymbol/s

Samples per symbol

RRC roll-off α0.35

RRC span per filter

Decision timing offset (UI)+0.00 UI

Echo amplitude |a|0.00

Echo delay τ (UI)0.50 UI

Echo phase θ (degrees)0°

Decision-reference SNR (dB)30 dB

Fixed seed preset

Ideal RC support: 13.500 kHz
RRC implementation: 65 taps
RMS EVM · R3: 3.43%
Decision errors: 0 / 240
Vertical opening proxy: 94.4%
Horizontal opening proxy: 0.74 UI

TX/RX/channel combined impulse

Finite calculation · real solid, imaginary dashed · center referenced to nominal decision time

D3 transmit spectrum

Finite Hann-windowed calculation · 19.531 Hz bins · ideal support separately marked

I-component eye at R3

24 representative two-UI overlays · 8 native samples/UI · sinc-interpolated display grid

Keyboard-selected overlaySelected trace begins at symbol 9; its nominal I sample is 0.7309.

Decision constellation at R3

First 96 included symbols · ideal squares, received circles, decision errors crossed

Active cause: Baseline: finite 8-symbol RRC filters, nominal timing, no echo, and 30 dB decision-reference SNR.

Counterfactual diagnosis · one selected cause removed at a time
Case	Changed assumption	RMS EVM	Symbol errors	Vertical opening	Horizontal opening
Active result	None	3.435%	0 / 240	94.4%	0.74 UI
Remove AWGN	Complex noise variance set to zero; timing and echo retained	1.611%	0 / 240	97.5%	0.74 UI
Restore timing	Decision phase set to 0 UI; noise and echo retained	3.435%	0 / 240	94.4%	0.74 UI
Remove echo	Echo amplitude set to zero; noise and timing retained	3.435%	0 / 240	94.4%	0.74 UI

Inspect impulse, eye, and constellation data

Finite TX/RX combined impulse · no channel · center normalized to 1
Offset	Amplitude
-8 UI	5.218885e-7
-7 UI	-7.255901e-5
-6 UI	8.662251e-4
-5 UI	-2.968087e-3
-4 UI	1.049110e-2
-3 UI	-2.622700e-3
-2 UI	7.167695e-4
-1 UI	1.256661e-4
+0 UI	1.000000e+0
+1 UI	1.256661e-4
+2 UI	7.167695e-4
+3 UI	-2.622700e-3
+4 UI	1.049110e-2
+5 UI	-2.968087e-3
+6 UI	8.662251e-4
+7 UI	-7.255901e-5
+8 UI	5.218885e-7

TX RRC component pulse · representative symbol-spaced taps
Offset	Energy-normalized amplitude
-4 UI	7.224185e-4
-3 UI	-9.000550e-3
-2 UI	2.019768e-2
-1 UI	-2.994691e-2
+0 UI	3.874217e-1
+1 UI	-2.994691e-2
+2 UI	2.019768e-2
+3 UI	-9.000550e-3
+4 UI	7.224185e-4

Selected I-eye overlay · symbol 9
Phase	I amplitude
-1.00 UI	-0.688064
-0.75 UI	-0.248085
-0.50 UI	0.225423
-0.25 UI	0.590435
+0.00 UI	0.730915
+0.25 UI	0.602206
+0.50 UI	0.247222
+0.75 UI	-0.223164
+1.00 UI	-0.674960

First 16 included R3 decisions
k	Ideal I	Ideal Q	Observed I	Observed Q
8	-0.7071	-0.7071	-0.6881	-0.6935
9	0.7071	-0.7071	0.7309	-0.7153
10	-0.7071	-0.7071	-0.6750	-0.7099
11	-0.7071	-0.7071	-0.7336	-0.7372
12	0.7071	0.7071	0.6744	0.6877
13	0.7071	-0.7071	0.6773	-0.7447
14	0.7071	-0.7071	0.7068	-0.6975
15	-0.7071	-0.7071	-0.7366	-0.6999
16	0.7071	0.7071	0.7369	0.6887
17	-0.7071	-0.7071	-0.7121	-0.7053
18	0.7071	0.7071	0.6974	0.7434
19	-0.7071	0.7071	-0.7106	0.6949
20	0.7071	0.7071	0.6897	0.7118
21	0.7071	0.7071	0.7412	0.7285
22	-0.7071	0.7071	-0.7018	0.6991
23	0.7071	-0.7071	0.7211	-0.6816

Crossing distribution: Mean 0.502 UI, standard deviation 0.090 UI, n = 243 sign-changing component transitions.
Population: 240 of 256 symbols included; 8 removed from each burst edge (16 total).
Channel guard: 0 symbols; the 17-tap fractional-delay branch is bypassed exactly because echo amplitude is zero.
Noise variance: 1.0000e-3 complex variance; each I/Q component uses half, before the unit-energy RX filter.

Download deterministic CSV

hRRC(x) → upsample → TX FIR → [δ(t) + ae^jθδ(t−τ)] → complex AWGN → RX FIR → sampleEach finite RRC has unit energy. The active channel uses a DC-normalized 17-tap Blackman-windowed sinc for fractional delay and contributes eight implementation-delay samples, which are removed before R3 timing is declared. Positive timing offset samples later. AWGN has complex variance 10^(−SNR/10), so each I/Q component has half that variance; SNR is the unit-symbol-energy matched-filter reference, not a measured receiver C/N.

Model boundary. Deterministic LTI complex baseband; 256 PRBS-9-derived unit-average-energy symbols; finite selected RRC and channel FIRs; periodic-Hann D3 spectrum; raw R3 EVM with no fitted gain, phase, timing, or channel correction. Excluded population is first/last span + channel guard. No PA, RF filter, clock recovery, coding, fading model, measurement receiver, or general equalizer is implemented. Ideal RC support and the local finite-record 99% width are different quantities; neither is asserted as a formal occupied, necessary, channel, or compliance bandwidth.

Static workbench fallback

JavaScript is unavailable. The complete lesson, checked fixtures, and these default/fault summaries remain available.

10 ksymbol/s ideal raised-cosine support · calculated, not occupied bandwidth
Roll-off	Signed edge	Total null-to-null
0.10	±5.50 kHz	11.00 kHz
0.35	±6.75 kHz	13.50 kHz
1.00	±10.00 kHz	20.00 kHz

Default: RMS EVM 3.43 percent; 0 symbol errors in 240; vertical opening 94.4 percent of ideal; sample phase +0.00 UI.
+0.20 UI timing: RMS EVM 27.88 percent; 0 symbol errors in 240; vertical opening 49.8 percent of ideal; sample phase +0.20 UI.
0.30 echo at 0.50 UI: RMS EVM 27.41 percent; 0 symbol errors in 236; vertical opening 88.3 percent of ideal; sample phase +0.00 UI.
10 dB AWGN case: RMS EVM 30.77 percent; 1 symbol errors in 240; vertical opening 49.2 percent of ideal; sample phase +0.00 UI.

Default filter fixture: 65 taps per RRC, 32 samples = 4 UI = 0.4 ms delay per filter, 0.8 ms combined; center 1.0 and residual nonzero symbol-spaced energy 2.540729794e−4 (−35.950 dB).

Common misconceptionIf EVM is high, increasing SNR always fixes it.

SNR only controls the additive-noise term in this model. Finite truncation, wrong timing, and a two-path channel remain when AWGN is removed; the counterfactual table makes that separation explicit.

Deliberate boundary. The workbench implements no timing recovery, channel estimator, or equalizer. It shows the evidence those later designs would need; algorithms remain deferred.

Evidence Timing and channel/error interpretation: P02-S2 and P02-S9; deterministic model order, normalization, seed, and guards are printed in the workbench and regression tested.

10 / 10

Update the pulse-shaping decision

Which baseline meets the supplied width/residual target without hiding latency and recovery risk?

Use an illustrative internal target: ideal total RC support no wider than 15 kHz and finite no-channel symbol-spaced residual energy no greater than 1×10⁻³ (−30 dB), before assigning channel/noise margin. It is not a standard limit.

10 ksymbol/s candidate comparison · internal target only
Candidate	Ideal support	Finite sampled residual	Delay per RRC	Decision
α = 0.10, span 2	11.00 kHz · width passes	1.193e-1 (-9.23 dB) · fails	1 UI	Reject: misses residual target by 119.3×
α = 0.35, span 8	13.50 kHz · width passes	2.541e-4 (-35.95 dB) · passes	4 UI = 0.4 ms	Select as first-pass model baseline

Earlier record · source/rate: Illustrative 20.0 kbit/s information source; framing/coding remain unspecified. Gray QPSK maps two uncoded teaching bits/symbol at Rₛ = 10 ksymbol/s.
D2 mapping / normalization: Gray QPSK, E[|aₖ|²] = 1, fixed PRBS-9-derived 256-symbol teaching burst; seed preset `path02-default`.
Pulse implementation: RRC TX + equal RRC RX, α = 0.35, span 8 symbols per filter, 8 samples/symbol, 65 symmetric unit-energy taps each.
Bandwidth definition: Ideal combined RC signed complex-baseband support ±6.75 kHz, 13.5 kHz total null-to-null. Not relabeled occupied, necessary, channel, or measured bandwidth.
Delay / transients: 0.4 ms per RRC and 0.8 ms pair delay at 10 ksymbol/s. Metrics exclude first/last span + channelGuard symbols; default is 8 per edge, 240 included.
Planes: D2 mapper → D3 TX-filter output → deterministic baseband channel/AWGN → R3 matched-filter decision plane.
Decision / eye assumptions: Nominal phase 0 UI; positive timing is later. I-eye displays representative histories; proxies use 5th/95th percentile component separation and declared crossing population.
Channel / noise / EVM: Baseline echo off; 30 dB unit-symbol-energy matched-filter SNR. Raw RMS EVM uses included R3 symbols with no fitted gain, phase, timing, or equalizer correction.
Finite fixture: Combined center 1; nonzero symbol-spaced residual energy 2.540729794e−4 (−35.950 dB) across ±8 UI.
Unresolved evidence: Actual channel impulse/fading, clock acquisition/tracking, equalizer need, burst guard budget, PA/RF filters, coding, measurement procedure, and applicable standard limits.

Handoff to 02.7

The pulse, reference planes, timing, finite-filter error, and channel assumptions are now explicit. 02.7 combines EVM, spectrum, peaks, and impairment evidence into the wider waveform trade study.

Evidence Baseline values from Portfolio Conventions v1.0; candidate rejection and finite fixtures from pulse-eye/1.0 equation-level tests. The width/residual target is illustrative.

Ungraded review

Check your understanding

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

01Write the pulse-amplitude waveform built from symbols aₖ and a pulse p(t).
Model answer
s(t) = Σₖ aₖp(t − kTₛ). The equation also needs the symbol interval Tₛ, pulse definition and scaling, and the reference plane where s(t) is observed.
02How can pulses overlap while the combined response still has zero ISI?
Model answer
Zero ISI constrains the combined TX/channel/RX response only at the declared decision instants: g(0) is nonzero and g(nTₛ) = 0 for every nonzero integer n under correct timing. Between those samples, neighboring pulses may overlap strongly.
03For Rₛ = 10 ksymbol/s and α = 0.35, what are the ideal RC support and signed edges?
Model answer
B = (1 + α)Rₛ = 13.5 kHz total null-to-null, with signed complex-baseband edges at ±6.75 kHz. This is ideal support, not automatically occupied, necessary, channel, or measurement bandwidth.
04An 8-symbol RRC runs at 8 samples/symbol. Find taps and delay for one filter and the TX/RX pair at 10 ksymbol/s.
Model answer
Tap count = 8×8 + 1 = 65. A symmetric 65-tap FIR delays 32 samples = 4 symbols = 0.4 ms. Two such filters contribute 64 samples = 8 symbols = 0.8 ms before other latency.
05Noise falls but the eye remains history-dependent and asymmetric. What should you investigate first?
Model answer
Investigate deterministic channel or filter ISI and the sampling phase. Lowering AWGN only narrows random spread; it does not remove a delayed replica, truncation residual, or wrong decision instant.
06What must the v6 record state beside an eye or EVM number?
Model answer
Plane, pulse and finite filters, symbol/sample rates, timing reference, channel and noise assumptions, scaling, seed, excluded burst edges, included population, metric definition, and whether any gain/phase/timing/channel correction was applied.

Sources and further study

The lesson paraphrases stable theory and redraws every figure from equations. Live links were checked on 5 September 2026. ITU-R SM.328-12 (09/2025) remains in force; R&S 3683.8038.52 is Version 01.00 (October 2022). No output on this page is measured evidence.

Primary technical references

P02-S2 · Informative: J. G. Proakis and M. Salehi, Digital Communications, 5th ed.; Chapters 4–5 for receiver/timing foundations and Chapter 9 for band-limited signaling and pulse shaping.
P02-S9 · Informative: B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed., Chapters 1–4.
P02-S1 · Informative: A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed., Chapters 3–7.
P02-S3 · Informative: MIT OpenCourseWare: Unified Engineering Signals and Systems.
P02-S8 · View orientation only: Rohde & Schwarz, Understanding EVM, Version 01.00.
P02-S4 · Normative terminology only: ITU-R SM.328-12, Spectra and bandwidth of emissions.

Evidence labels and model trace

Definition: PAM construction, combined Nyquist condition, RC/RRC split, raw R3 EVM, and eye proxies state their conventions beside use.
Derived: rectangular zeros, ideal RC support, tap count, delay, singular limits, and candidate comparison follow the displayed equations.
Simulated: pulse-eye/1.0 uses fixed PRBS-9-derived symbols, deterministic complex AWGN, finite FIRs, and a versioned fractional-delay channel.
Illustrative: the node, −35 dB filter proxy, 15 kHz/−30 dB selection target, and two-path cases are synthetic teaching requirements.
Measured: none. Measurement receiver setup and uncertainty belong to 08.5.

Implemented prerequisite02.1 · Time & Frequency Previous module02.5 · Digital Symbols Next module02.7 · Modulation Quality

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