Signals in Time

A signal is a physical quantity that varies with time and carries information. Before asking what frequencies are present in a signal, the first question to answer is: what does it look like, and how do we describe it precisely? This chapter builds the vocabulary and tools for answering that question. It introduces analog and digital signals, develops the harmonic signal as the fundamental building block of periodic waveforms, and gives careful attention to the concept of phase, including what it means for one signal to lead or lag another. The chapter closes with sampling and the Nyquist criterion, which govern how analog signals are converted to the discrete form that computers and digital systems require.

Learning Objectives:

• Distinguish between analog and digital signals and describe their key characteristics.
• Identify the four parameters of a harmonic signal: amplitude, frequency, period, and phase.
• Calculate peak, peak-to-peak, and RMS amplitude for a sinusoidal signal.
• Determine whether one sinusoidal signal leads or lags another, and express the phase difference in radians and degrees.
• Recognize square, sawtooth, and triangle waveforms and describe their defining features.
• State the Nyquist sampling criterion and apply it to determine a sufficient sampling rate.
• Explain what aliasing is and why anti-aliasing filters are necessary before analog-to-digital conversion.

What Is a Signal?

In electrical and computer engineering, a signal is a time-varying physical quantity that conveys information. Signals take many forms. Acoustic pressure waves carry speech and music. Mechanical vibration carries information about rotating machinery. Electromagnetic fields carry radio, radar, and optical communications over distances ranging from millimeters to billions of kilometers. Voltages and currents in circuits carry information between components on a printed circuit board and between instruments in a laboratory.

All of these are signals, and the mathematical tools developed in this chapter apply to all of them equally. The focus here is on voltage and current signals, because these are the quantities that circuits process directly and that the laboratory instruments in this course, including the oscilloscope and the M2K, measure. Understanding how to describe and analyze a voltage signal in the time domain transfers immediately to any other signal type.

Signals arise throughout engineering practice. A microphone converts the pressure variations of a speaker's voice into a voltage that follows the same time variation. An electrocardiogram (ECG) sensor converts the electrical activity of the heart muscle into a slowly varying voltage that a physician can read. A wireless antenna intercepts a propagating electromagnetic field and produces a tiny voltage from which a receiver recovers the encoded audio or data. In each case, the signal passes through one or more physical forms before reaching the circuit that processes it, but its mathematical character as a time-varying quantity remains the same throughout.

The most natural way to observe a signal is to plot its value as a function of time. This representation is called the time domain. For a voltage signal, it is the view that an oscilloscope provides: voltage on the vertical axis, time on the horizontal axis. The same representation applies to any other signal type: acoustic pressure, electric field strength, or optical intensity plotted against time all yield time-domain representations governed by the same mathematics. In this chapter, signals will be presented in the time domain using voltage as the representative quantity. The following chapter will introduce the complementary frequency domain, which reveals the sinusoidal components present in any signal.

Analog and Digital Signals

Analog Signals

Key Concept: Analog Signal

An analog signal is a continuous-time, continuous-valued electrical signal. Its value can be any real number within its range, and it changes smoothly without sudden jumps.

Analog signals closely mirror the physical phenomena they represent. Sound pressure varies continuously as a speaker's mouth opens and closes; the corresponding microphone voltage varies in the same continuous way. Temperature rises and falls gradually over the course of a day; a temperature sensor produces a smoothly changing voltage that follows it.

Several characteristics define analog signals in engineering practice:

1. Continuity in time: The signal is defined at every instant. There are no gaps or undefined moments.
2. Continuity in value: The signal can take any value in its operating range, not merely a fixed set of levels.
3. Faithful representation of physical phenomena: Many quantities in nature are inherently analog. Analog signals preserve the full detail of these quantities.
4. Susceptibility to noise: Because any value is meaningful, any small disturbance added to an analog signal is indistinguishable from real information. Noise corrupts analog signals in a way that is difficult to reverse.

Figure 1: An analog voltage signal representing human speech. The signal varies smoothly and continuously, with no abrupt jumps. Source: Ordaz et al., Journal of Applied Research and Technology, 10(5), 783–790, 2012.

Digital Signals

Key Concept: Digital Signal

A digital signal is a signal that takes only a finite set of discrete values, typically two voltage levels representing the binary digits 0 and 1.

Rather than following the continuous variations of a physical quantity directly, a digital signal represents information as a sequence of binary values. A high voltage level (for example, 3.3 V) represents a logical 1; a low voltage level (for example, 0 V) represents a logical 0. All information, including text, audio, images, and video, can be encoded as a sequence of these two values.

Figure 2: A digital signal representing binary data. Only two voltage levels are used. Transitions between them are nearly instantaneous.

Digital signals offer three important advantages over analog signals:

1. Noise immunity: A circuit reading a digital signal only needs to decide whether the voltage is closer to the high level or the low level. Small amounts of noise do not change that decision, and the signal can be regenerated perfectly at each stage of a long transmission link.
2. Exact reproduction: A digital file copied from one medium to another is bit-for-bit identical to the original. Analog recordings degrade with each copy.
3. Computational compatibility: Processors, memory, and communication networks operate in the digital domain. Storing or processing a signal digitally places it directly in the form that computers require.

The Bridge Between Analog and Digital

The physical world is analog; computation is digital. Most practical systems contain both. An analog-to-digital converter (ADC) samples an analog signal at regular time intervals and represents each sample as a binary number. A digital-to-analog converter (DAC) reconstructs an analog signal from a sequence of binary values.

A smartphone illustrates the cycle: the microphone produces an analog voltage, the ADC converts it to digital data, a processor compresses and transmits it, the receiving device's DAC reconstructs an analog voltage, and the loudspeaker converts that voltage back to sound pressure. The conversion process introduces constraints that are explored in the Sampling and the Nyquist Criterion section below.

The Harmonic Signal

The Fundamental Building Block

The simplest periodic signal is the harmonic signal: a pure cosine or sine wave at a single frequency. It may seem overly simple, but it is the most important signal in engineering and physics. The reason is a remarkable mathematical result established by Joseph Fourier in the early nineteenth century: any periodic signal, regardless of its shape, can be represented exactly as a sum of harmonic signals. This means that understanding how a circuit responds to a single cosine at a given frequency is sufficient to predict how it will respond to any periodic signal whatsoever. The harmonic signal is the atom from which all other signals are built.

Mathematical Description

A harmonic signal is written as:

$$v(t) = V_o \cos(\omega t + \phi) = V_o \cos(2\pi f t + \phi)$$

where:

• $v(t)$ is the instantaneous voltage at time $t$,
• $V_o > 0$ is the peak amplitude (in volts),
• $\omega$ is the angular frequency (in radians per second),
• $f$ is the frequency (in hertz),
• $\phi$ is the phase (in radians or degrees).

Figure 3: A harmonic signal $v(t) = V_o \cos(2\pi f t)$ showing the three amplitude measures. The peak amplitude $V_o$ is the maximum value. The peak-to-peak amplitude $V_{pp} = 2V_o$ is the full span from minimum to maximum. The RMS amplitude $V_\text{RMS} = V_o/\sqrt{2}$ (dashed line) is the effective value used in power calculations. The period $T = 1/f$ is the duration of one complete cycle.

Amplitude

The amplitude of a harmonic signal describes the size of its oscillation. Three measures are commonly used:

1. Peak amplitude $V_o$: The maximum value the signal reaches. The signal ranges from $-V_o$ to $+V_o$.
2. Peak-to-peak amplitude $V_{pp}$: The total span from the most negative value to the most positive value. For a cosine, $V_{pp} = 2V_o$. This is the measurement most directly read from an oscilloscope display.
3. Root mean square (RMS) amplitude $V_\text{RMS}$: The effective value of the signal for power calculations. It is defined as:

$$V_\text{RMS} = \sqrt{\frac{1}{T}\int_0^T v^2(t)\, dt}$$

For a cosine, $V_\text{RMS} = V_o / \sqrt{2}$ (derived below). Household mains voltage is specified as an RMS value: 120 V RMS in North America corresponds to a peak amplitude of approximately 170 V.

Worked Example: RMS Amplitude of a Cosine

Problem: Find $V_\text{RMS}$ for $v(t) = V_o \cos(\omega t + \phi)$.

Solution:

$$v^2(t) = V_o^2 \cos^2(\omega t + \phi) = \frac{V_o^2}{2}\bigl[1 + \cos(2\omega t + 2\phi)\bigr]$$

Integrating over one full period $T = 2\pi/\omega$:

$$\frac{1}{T}\int_0^T \cos(2\omega t + 2\phi)\, dt = 0$$

because a complete cosine integrates to zero. Therefore:

$$V_\text{RMS} = \sqrt{\frac{1}{T}\int_0^T v^2(t)\, dt} = \sqrt{\frac{V_o^2}{2}} = \frac{V_o}{\sqrt{2}}$$

Frequency and Period

The frequency $f$ specifies how many complete cycles the signal completes per second. Its unit is the hertz (Hz), defined as one cycle per second. The period $T$ is the duration of one complete cycle:

$$T = \frac{1}{f}$$

A 1 kHz signal completes 1000 cycles every second; its period is $T = 1$ ms. A 60 Hz mains signal has a period of approximately 16.7 ms.

The angular frequency $\omega$ measures the same rate of oscillation but in radians per second rather than cycles per second. One complete cycle covers $2\pi$ radians, so:

$$\omega = 2\pi f$$

Angular frequency appears naturally in the mathematics of circuits containing capacitors and inductors, and will be used extensively in the AC Analysis chapter.

Phase

The phase $\phi$ specifies the position of the cosine waveform relative to the reference time $t = 0$. A signal with $\phi = 0$ has its first positive peak at $t = 0$. A nonzero phase shifts the waveform in time:

• A positive phase ($\phi > 0$) moves the peak to the left, meaning the peak occurs before $t = 0$.
• A negative phase ($\phi < 0$) moves the peak to the right, meaning the peak occurs after $t = 0$.

Phase is expressed in radians or degrees. The conversion is $\phi_\text{deg} = \phi_\text{rad} \times (180/\pi)$.

Figure 4: Three harmonic signals at the same frequency and amplitude, differing only in phase. The blue curve has $\phi = +\pi/3$ and its peak arrives before the reference (black) curve: it leads by $60^\circ$. The red curve has $\phi = -\pi/3$ and its peak arrives after the reference: it lags by $60^\circ$.

Leading and Lagging Sinusoids

Phase as Time Shift

Two sinusoidal signals at the same frequency may differ by a phase offset. Consider:

$$v_A(t) = V_A \cos(\omega t + \phi_A)$$

$$v_B(t) = V_B \cos(\omega t + \phi_B)$$

The phase difference between them is $\phi_A - \phi_B$. This difference corresponds directly to a time shift. If the peak of $v_A$ occurs at time $t_A$ and the peak of $v_B$ occurs at time $t_B$, then:

$$\phi_A - \phi_B = \omega(t_B - t_A)$$

A positive phase difference means $v_A$ reaches its peak earlier in time than $v_B$.

Which Signal Leads?

Rule

The signal whose peak arrives first (earlier in time) is said to lead. The signal whose peak arrives later is said to lag.

If $\phi_A > \phi_B$, then $v_A$ leads $v_B$ by $(\phi_A - \phi_B)$ radians. Equivalently, $v_B$ lags $v_A$ by the same amount.

A common point of confusion is the apparent paradox: a positive phase looks like the signal has been shifted to the left on the time axis, yet the convention is that a positive phase means the signal appears earlier. Both statements describe the same fact. Reading a time-domain plot from left to right corresponds to moving forward in time. A peak that appears to the left of another peak on the plot is one that occurs at a smaller value of $t$, meaning it arrives first. The signal with the leftward-shifted (earlier) peak is the one that leads.

Worked Example: Identifying Lead and Lag

Given:

$$v(t) = 5\cos\!\left(1000t + \tfrac{\pi}{4}\right) \text{ V}, \qquad i(t) = 2\cos\!\left(1000t - \tfrac{\pi}{4}\right) \text{ A}$$

Questions: (a) What is the phase difference? (b) Which quantity leads? (c) Express the phase difference in degrees.

Solution:

1. The phase of $v$ is $\phi_v = +\pi/4$ and the phase of $i$ is $\phi_i = -\pi/4$. The phase difference is:

$$\phi_v - \phi_i = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2} \text{ rad}$$

1. Since $\phi_v > \phi_i$, the voltage $v(t)$ leads the current $i(t)$ by $\pi/2$ radians. Equivalently, $i(t)$ lags $v(t)$ by $\pi/2$ radians.

2. Converting: $\pi/2 \times (180^\circ/\pi) = 90^\circ$.

Why Phase Matters in Circuits

Phase relationships between voltage and current carry physical meaning. When a sinusoidal source drives a resistor, the current and voltage are in phase: their peaks and zero crossings coincide. The resistor absorbs power continuously.

When a sinusoidal source drives a capacitor or inductor, the current and voltage are out of phase by $90^\circ$. For a capacitor, the current leads the voltage by $90^\circ$. For an inductor, the current lags the voltage by $90^\circ$. A $90^\circ$ phase difference means the device alternately absorbs energy from the circuit and returns it, rather than dissipating it as heat. This distinction between resistive (in-phase) and reactive (out-of-phase) behavior is central to the analysis of AC circuits and will be developed fully in the AC Analysis chapter.

Note

A mnemonic used in introductory courses is ELI the ICE man. In an inductor (L), voltage (E) leads current (I): ELI. In a capacitor (C), current (I) leads voltage (E): ICE. While mnemonics are a starting point, the physical reasoning above is more reliable for unfamiliar situations.

Common Periodic Waveforms

A periodic signal repeats the same pattern at regular intervals. The time required for one complete repetition is the period $T$, and the fundamental frequency is $f_0 = 1/T$. The harmonic cosine is the simplest periodic signal, but three other waveforms appear constantly in electrical engineering practice and in the laboratory.

Square Wave

A square wave alternates between two values, spending equal time at each. When the high duration equals the low duration, the waveform has a duty cycle of 50%.

Figure 5: A square wave with 50% duty cycle, period $T$, and peak amplitude $V_o$. The sharp vertical transitions require high-frequency content to be reproduced faithfully.

The sharp vertical edges of a square wave are impossible for a physical circuit with limited bandwidth to reproduce exactly. In practice, square wave edges are slightly rounded. Digital clock signals, pulse-width modulation (PWM) in motor controllers, and logic-level outputs from microcontrollers are all examples of square-wave signals.

Sawtooth Wave

A sawtooth wave rises linearly from its minimum to its maximum over each period, then drops abruptly back to the minimum.

Figure 6: A sawtooth wave with period $T$ and peak amplitude $V_o$. The signal rises linearly from $-V_o$ to $V_o$ over each period, then resets abruptly to $-V_o$. Function generators and oscilloscope timing circuits use sawtooth signals internally.

Triangle Wave

A triangle wave rises linearly to its maximum and then falls linearly back to its minimum, with no abrupt transitions.

Figure 7: A triangle wave with period $T$. The signal rises linearly from $-V_o$ at $t = 0$ to its peak $V_o$ at $T/4$, falls back to $-V_o$ at $T/2$, and repeats. Unlike the square and sawtooth waves, the triangle wave has no abrupt transitions.

Connecting Waveform Shape to Frequency Content

A key observation to carry into the following chapter is that the sharpness of a waveform's features is directly related to the high-frequency content required to represent it.

• The square wave has instantaneous (vertical) transitions. Reproducing them requires contributions from harmonics extending to very high frequencies.
• The sawtooth wave also has an abrupt reset, but its ramp is gradual. It requires a wide range of harmonics, though the very high harmonics are less prominent than in a square wave.
• The triangle wave has no abrupt transitions. Its corners are sharp but finite in slope. The triangle wave's high-frequency harmonics decrease much more rapidly with frequency than those of the square or sawtooth.
• The cosine has no sharp features at all. It is perfectly smooth and contains exactly one frequency.

This pattern — smoother waveforms need fewer high-frequency components — will be made quantitative when Fourier Series are introduced in the following chapter.

Worked Example: RMS Amplitude of a Rectangular Wave

A rectangular wave has peak amplitude $V_o$, period $T$, and duty cycle $D$, meaning it is high for a fraction $D$ of each period and zero otherwise. Find $V_\text{RMS}$ for $D = 0.25$.

Solution:

Over one period, the signal equals $V_o$ for duration $DT$ and $0$ for the remaining $(1-D)T$. The mean square value is therefore:

$$\frac{1}{T}\int_0^T v^2(t)\,dt = \frac{1}{T}\left[\int_0^{DT} V_o^2\,dt + \int_{DT}^{T} 0^2\,dt\right] = D V_o^2$$

Taking the square root:

$$V_\text{RMS} = V_o\sqrt{D}$$

For $D = 0.25$:

$$V_\text{RMS} = V_o\sqrt{0.25} = \frac{V_o}{2}$$

Interpretation: a 25% duty cycle rectangular wave delivers the same power to a resistor as a DC voltage of $V_o/2$. Compare this with a cosine of the same peak amplitude, which gives $V_\text{RMS} = V_o/\sqrt{2} \approx 0.707\,V_o$. The rectangular wave at 25% duty cycle delivers less power because it is at zero for three quarters of each cycle.

Sampling and the Nyquist Criterion

From Continuous to Discrete: What Sampling Means

An analog-to-digital converter does not capture the complete continuous waveform of a signal. It takes samples: measurements of the signal's instantaneous voltage at equally spaced moments in time. The time between samples is the sampling period $T_s$, and its reciprocal is the sampling frequency (or sampling rate) $f_s$:

$$f_s = \frac{1}{T_s}$$

Each sample is converted to a binary number. The collection of these numbers is the digital representation of the signal.

The question that arises immediately is: how rapidly must the ADC sample the signal to capture it faithfully? If the sampling rate is too low, the recorded sequence of numbers no longer accurately represents the original signal.

Figure 8: Left: a 1 kHz cosine sampled at 5 kHz (five samples per cycle). The samples (blue) capture the waveform faithfully. Right: the same cosine sampled at 1.2 kHz (fewer than two samples per cycle). The sparse red samples are consistent with a much lower frequency, which is an alias of the original.

The Nyquist Sampling Criterion

The minimum sampling rate required to represent a signal without distortion is determined by the Nyquist criterion, established by Harry Nyquist in the 1920s.

Nyquist Sampling Criterion

To reconstruct an analog signal from its samples without error, the sampling frequency $f_s$ must be at least twice the highest frequency $f_\text{max}$ present in the signal:

$$f_s \geq 2\, f_\text{max}$$

The value $f_s / 2$ is called the Nyquist frequency.

The factor of 2 has an intuitive justification. A sinusoid at frequency $f_0$ has exactly one peak and one trough per cycle. To distinguish a peak from a trough, at minimum one sample must fall near the peak and one near the trough. That requires at least two samples per cycle, which corresponds to $f_s \geq 2f_0$. Fewer than two samples per cycle leaves the distinction between peaks and troughs ambiguous.

Aliasing: When Sampling Goes Wrong

When the sampling rate falls below the Nyquist criterion, a phenomenon called aliasing occurs. High-frequency components of the signal are indistinguishable from low-frequency components at the sampled rate, and they appear in the reconstructed signal at incorrect, lower frequencies.

The Core Problem: One Set of Samples, Two Possible Signals

Sampling records only the signal's value at discrete moments. Between those moments, the original waveform is discarded. This raises a fundamental question: given only the recorded sample values, can the original signal always be recovered uniquely?

The answer is no, unless the Nyquist criterion is satisfied. When too few samples are taken, multiple different sinusoids can produce exactly the same sequence of sample values. The figure below shows this directly.

Figure 9: Two cosines at different frequencies, 1 Hz (dashed red) and 3 Hz (solid black), produce identical sample values (blue dots) when sampled at $f_s = 4$ Hz. Given only the blue dots, it is impossible to tell which signal was the source. The Nyquist criterion is violated because $f_s = 4$ Hz is less than $2 \times 3$ Hz.

The two signals above are called aliases of each other at the given sampling rate. When the digital system reconstructs a signal from the samples, it will always recover the lower-frequency alias, because that is the simplest signal consistent with the data.

Calculating the Alias Frequency

The alias frequency is determined precisely by the sampling rate and the original signal frequency. When a signal at frequency $f$ is sampled at rate $f_s$ and the Nyquist criterion is violated, the alias appears at:

$$f_\text{alias} = \left| f - n \cdot f_s \right|$$

where $n$ is the positive integer that brings the result into the range $[0,\ f_s/2]$.

Worked Example: Finding the Alias Frequency

Example 1. Verify the result in Figure 9: $f = 3$ Hz, $f_s = 4$ Hz.

$$n = 1: \quad f_\text{alias} = |3 - 1 \times 4| = 1\,\text{Hz}$$

The result, 1 Hz, lies in $[0,\ 2]$ Hz. The 3 Hz signal aliases to 1 Hz.

Example 2. A 900 Hz signal is sampled at $f_s = 1000$ Hz.

$$n = 1: \quad f_\text{alias} = |900 - 1 \times 1000| = 100\,\text{Hz}$$

The result, 100 Hz, lies in $[0,\ 500]$ Hz. The 900 Hz signal aliases to 100 Hz. See Figure 10 below.

Figure 10: A 900 Hz signal (solid black) sampled at $f_s = 1$ kHz. The Nyquist criterion requires $f_s \geq 1800$ Hz; at 1 kHz this is violated. The alias frequency is $|900 - 1000| = 100$ Hz (dashed red). Every blue sample dot lies exactly on both curves: the 900 Hz signal and the 100 Hz alias produce identical sample values. The ADC cannot distinguish between them, and the original 900 Hz information is irrecoverably lost.

An Everyday Analogy: The Wagon-Wheel Effect

In film and video, a spinning wheel can appear to rotate slowly, stand still, or even turn backwards. The camera samples the wheel's angular position at the frame rate, typically 24 or 30 frames per second. If the wheel completes nearly one full revolution between frames, the camera records it at nearly the same angular position each time. The sampled positions are consistent with a very slowly rotating wheel, which is the alias. The true high rotation rate is lost.

The parallel with electrical signals is exact. The camera frame rate corresponds to $f_s$. The wheel's rotation rate corresponds to the signal frequency $f$. The apparent slow rotation is the alias at $f_\text{alias}$. The only solution, in both cases, is to sample fast enough that the ambiguity cannot arise.

Anti-Aliasing Filters

The solution to aliasing is to prevent high-frequency signal components from reaching the ADC in the first place. An anti-aliasing filter, placed before the ADC in the signal chain, is a low-pass filter that attenuates all frequency content above $f_s / 2$ before sampling occurs. Once a component has been aliased, it cannot be removed from the digital data; the filter must act on the analog signal before conversion.

The design of low-pass filters is covered in the Inductors, Capacitors, and Filters chapter. For now, the important point is that any real ADC system includes an anti-aliasing filter as a necessary component of its design.

Practical Sampling Rates

The table below lists sampling rates used in several common applications, together with the signal bandwidth they must capture and the resulting Nyquist margin.

Note that practical systems sample somewhat above the Nyquist minimum. This margin accommodates the fact that real anti-aliasing filters do not cut off infinitely sharply at $f_s / 2$; a small guard band reduces the demand on the filter.

Quantization

Sampling discretizes the time axis. An additional process called quantization discretizes the amplitude axis. Each sample is represented as a binary integer with a fixed number of bits. An $n$-bit ADC divides the full-scale voltage range into $2^n$ levels. A 12-bit ADC (used in the M2K) produces $2^{12} = 4096$ levels. The difference between adjacent levels is the least significant bit (LSB) voltage, and any sample value that falls between two levels is rounded to the nearest one. This rounding introduces a small error called quantization noise.

For the M2K operating over a $\pm 5$ V range, the LSB voltage is approximately $10\text{ V} / 4096 \approx 2.4$ mV. Voltage differences smaller than this cannot be resolved by the instrument. This connects directly to the concept of resolution introduced in Lab 2.

Chapter Summary

This chapter introduced the vocabulary and tools for describing electrical signals in the time domain.

Analog signals vary continuously in time and value. Digital signals take only two values and represent information as binary sequences. The ADC and DAC convert between these two representations.

A harmonic signal $v(t) = V_o \cos(\omega t + \phi)$ is characterized by four parameters. The peak amplitude $V_o$ describes the signal's size. The angular frequency $\omega = 2\pi f$ and period $T = 1/f$ describe how rapidly it oscillates. The phase $\phi$ describes the position of its peak relative to $t = 0$. The RMS amplitude, $V_\text{RMS} = V_o / \sqrt{2}$, is the effective value for power calculations.

When two sinusoids at the same frequency differ in phase, the one whose peak arrives first leads; the other lags. Phase relationships between voltage and current determine whether a circuit element dissipates energy (resistor, in phase) or stores and returns it (capacitor or inductor, $90^\circ$ out of phase).

Square, sawtooth, and triangle waves are common periodic waveforms. Their sharp features signal the presence of high-frequency harmonics. The smoother the waveform, the more rapidly its harmonics diminish with increasing frequency.

The Nyquist criterion requires that the sampling rate satisfy $f_s \geq 2 f_\text{max}$ to avoid aliasing. Violating this criterion causes high-frequency components to masquerade as low-frequency ones in the digitized data, an error that cannot be corrected after the fact. Anti-aliasing filters remove signal content above $f_s/2$ before the ADC converts the signal.

Key formulas: