Peak values, amplitude, offset & period
Turn a waveform into quantitative descriptors without abusing the word amplitude. Define maxima/minima and peak-to-peak value, distinguish vertical offset from time average, derive the mean of a periodic signal, define periodicity and fundamental period rigorously, handle constant-signal edge cases, and separate waveform shape from vertical/time shifts.
- Read and define maximum, minimum and peak-to-peak values with the same physical units as the signal.
- Explain why the bare word amplitude can be ambiguous for a general waveform and use explicit descriptors such as peak, peak-to-peak or sinusoidal amplitude.
- Represent a vertical offset as addition of a constant and distinguish that operation from the time average of an arbitrary waveform.
- Define the average of a periodic signal over one period and show when a sinusoidal offset equals the period average.
- Define periodicity by x(t+T)=x(t) and distinguish any valid period from the fundamental period.
- Explain the special case of a constant function: every positive T is a period, so there is no unique smallest positive fundamental period.
- Analyze time shifts x(t−t0) and vertical shifts x(t)+C without changing waveform shape.
- Use dimensional and limiting checks to validate amplitude/period calculations before moving to frequency and phase.
A waveform swings between 1 V and 5 V. Is its “amplitude” 5 V, 4 V or 2 V?
All three numbers can appear in informal engineering speech, but they describe different quantities: maximum value, peak-to-peak excursion and half the excursion. This chapter replaces the bare word amplitude by explicit definitions, then separates vertical offset, time average and period.
02.1Start with quantities that are unambiguous: maximum, minimum and peak-to-peak
For a bounded real waveform x(t) over a stated interval or over one period, define
All three have the same physical unit as x. For a voltage ranging from 1 V to 5 V:
The subscript pp means peak-to-peak. It measures vertical excursion; it says nothing about how long the waveform spends near either extreme.
02.2The word amplitude needs a reference
For a general waveform there is no single universal number called “the amplitude” unless a convention is stated. Useful explicit quantities include:
For a pure sinusoid written later as x(t)=X0+A cos(...) with A≥0, the symbol A will have a precise meaning: sinusoidal peak amplitude measured from its center value X0. Do not transfer that definition blindly to an arbitrary pulse or asymmetric waveform.
02.3A vertical offset is a translation, not automatically an average
Adding a constant C shifts every value vertically:
Then ymax=xmax+C and ymin=xmin+C, while the peak-to-peak value is unchanged:
This is what a mathematical vertical offset means: a translation of the entire waveform. The midpoint between extrema also shifts by C.
But the midpoint between extrema is not generally the same as the time average. They coincide for many symmetric waveforms, including an ideal sinusoid around its center, but not for every waveform.
02.4Time average weighs values by how long the signal occupies them
Figure 2.1 — Vertical levels and time repetition describe different properties
The one question is: why is the midpoint of 1 V and 5 V equal to 3 V while this waveform's time average is 2 V?
The pattern repeats every 4 ms. It stays at 5 V for 1 ms and at 1 V for 3 ms. Thus x_max=5 V, x_min=1 V and x_pp=4 V, but the waveform spends unequal time at the two levels.
For a periodic signal with period T, define its average over one period as
For a genuinely periodic signal the result does not depend on which complete period is chosen as the integration interval.
In Figure 2.1, over one 4 ms period the signal is 5 V for 1 ms and 1 V for 3 ms:
Yet the midpoint of the extrema is
So “center between peaks” and “time average” are different definitions. For the sinusoid studied later, the symmetry makes the constant offset equal to the period average; that fact will be derived rather than assumed for all waveforms.
02.5Periodicity is a property of the whole function
A continuous-time signal x(t) is periodic if there exists a positive T such that
Any positive T satisfying the relation is a period. If a smallest positive period exists, it is the fundamental period, written here T0.
If T0=4 ms, then 8 ms, 12 ms and every positive integer multiple of 4 ms are also periods. They are not fundamental.
02.6A repeated-looking fragment is not enough: the equality must hold for all time
A finite data record can contain two similar-looking pulses without proving the underlying signal is periodic. Periodicity is a statement about the mathematical signal over its domain.
Likewise, a waveform that repeats for ten cycles and then changes is not periodic over all time under that full model. It may be described as a finite burst whose interior contains an approximately periodic carrier.
This distinction matters in measurement: an oscilloscope window can show only a finite interval. The observed repetition supports a model; it does not logically prove infinite repetition outside the recorded window.
02.7Time shifts and vertical shifts preserve shape but move it in different directions
A vertical shift by C is
A delay by a positive time t0 is
The minus sign can feel counterintuitive. Check one landmark: if x(t) has a peak at t=0, then y(t)=x(t−2 ms) has that peak when t−2 ms=0, hence at t=2 ms. The waveform moved to the right.
For a periodic signal, a time shift does not change maximum, minimum, peak-to-peak value or fundamental period. A vertical shift changes maximum, minimum and mean by C but does not change peak-to-peak value or period.
02.8Worked examples and independent checks
Example A — symmetric waveform. A waveform ranges from −2 V to +6 V. Then
If the waveform is symmetric in time about 2 V, its period average is also 2 V. Symmetry is the extra reason; extrema alone are not enough.
Example B — unequal dwell time. Figure 2.1 has the same half-excursion 2 V around the extrema midpoint 3 V, but its mean is only 2 V because it spends three quarters of each period at the lower level.
Example C — delay. If a 2 ms-wide pulse occupies 0≤t<2 ms, then replacing x(t) by x(t−5 ms) moves the same pulse to 5≤t<7 ms.
02.9Mastery exercises
- A voltage varies between −1 V and 7 V. Find maximum, minimum, peak-to-peak value, midpoint and half peak-to-peak excursion.
- A periodic two-level waveform is 8 V for 2 ms and 2 V for 6 ms. Find its period, extrema midpoint and time average.
- Why is the answer to “what is the amplitude?” incomplete for the waveform in exercise 2 unless the requested amplitude convention is stated?
- If y(t)=x(t)+3 V, which of xmax, xmin, xpp, period and average change, and by how much?
- If y(t)=x(t−4 ms), where does a landmark originally at t=1 ms appear?
- A signal satisfies x(t+2 ms)=x(t) for all t and no smaller positive period exists. List three other valid periods.
- Is a constant function periodic? Does it possess a unique fundamental period?
- An oscilloscope shows 20 nearly identical cycles. What evidence supports periodicity, and what does the finite record fail to prove?
Solutions
1. xmax=7 V, xmin=−1 V, xpp=8 V, xmid=3 V, half excursion=4 V.
2. T=8 ms. Midpoint=(8+2)/2=5 V. Average=[8·2+2·6]/8=3.5 V.
3. “Amplitude” might mean maximum magnitude, peak relative to a center, half peak-to-peak, or another convention. State the descriptor explicitly.
4. ymax=xmax+3 V, ymin=xmin+3 V, ypp=xpp, period unchanged, and the period average increases by 3 V.
5. At t=5 ms.
6. 4 ms, 6 ms and 8 ms, for example.
7. It satisfies the periodicity equality for every positive T, but has no unique smallest positive fundamental period.
8. The repeated record supports a periodic model over the observation window. It does not prove that the physical signal repeats indefinitely outside that window.
Chapter 03 will convert period into frequency and then define phase as a relative time-position parameter rather than a mysterious angle attached to a waveform.