Signal Spectrum, Fourier Analysis, Bandwidth, Filters, and Noise

Comprehensive practice flashcards reviewing signal spectrums, US frequency allocations, Fourier series and transforms, filter circuits, bandwidth definitions, and noise analysis.

Fundamental Frequency ($f_0$)

The lowest frequency component of a periodic waveform, defined as the reciprocal of the fundamental time period ($f_0 = \frac{1}{T_0}$).

Modulation

A technique used to position individual voice or data channels next to each other within the frequency spectrum to facilitate simultaneous transmission across transmission methods such as AM, FM, or Satellite.

Extremely Low Frequency (ELF) Band

The radio spectrum band spanning $3\,\text{Hz}$ to $3\,\text{kHz}$, utilized for submarine communication due to deep-water penetration, geophysical sensing, and limited military communications.

Medium Frequency (MF) Band

The radio spectrum band spanning $300\,\text{kHz}$ to $3\,\text{MHz}$, used for AM radio broadcasting ($530\,\text{kHz}$ to $1700\,\text{kHz}$), maritime communication, and aviation non-directional beacons (NDBs).

High Frequency (HF) Band

The radio spectrum band spanning $3\,\text{MHz}$ to $30\,\text{MHz}$, used for shortwave radio broadcasting, maritime/aviation communications, and military beyond-line-of-sight communication via ionospheric reflection.

Very High Frequency (VHF) Band

The radio spectrum band spanning $30\,\text{MHz}$ to $300\,\text{MHz}$, allocated for FM radio ($88\,\text{MHz}$ to $108\,\text{MHz}$), air traffic control ($118\,\text{MHz}$ to $137\,\text{MHz}$), marine radio, and emergency services (police, fire, EMS).

Ultra High Frequency (UHF) Band

The radio spectrum band spanning $300\,\text{MHz}$ to $3\,\text{GHz}$, used for cellular networks (4G, low-band 5G), digital TV broadcasting, $2.4\,\text{GHz}$ Wi-Fi, GPS (L1 = $1.575\,\text{GHz}$), and public safety radios.

Super High Frequency (SHF) Band

The radio spectrum band spanning $3\,\text{GHz}$ to $30\,\text{GHz}$, used for 5G cellular (mid-band and mmWave), Wi-Fi ($5\,\text{GHz}$ and $6\,\text{GHz}$ Wi-Fi 6E/7), Ku/Ka band satellite communications, radar systems, and microwave backhaul links.

Extremely High Frequency (EHF) Band

The radio spectrum band spanning $30\,\text{GHz}$ to $300\,\text{GHz}$, used for advanced 5G mmWave ($24\,\text{GHz}$ to $40\,\text{GHz}$), high-resolution radar, scientific sensing (atmospheric and astronomy), and experimental high-data-rate links.

Fourier Series

A mathematical representation that expresses a periodic time-domain signal as the sum of a constant DC value, cosine harmonic terms, and sine harmonic terms.

Fourier Series DC Value ($a_0$)

The average value of a periodic signal $x(t)$ over one period $T_0$, determined by the area under the curve per unit time: $a_0 = \frac{1}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t)\,dt$.

Fourier Transform

An integral transform that converts a continuous time-domain signal $x(t)$ into its frequency-domain spectrum $X(f)$, defined by $X(f) = \int_{-\infty}^{\infty} x(t)e^{-j2\pi ft}\,dt$.

Fourier Transform of a Cosine Wave

The spectral representation of $x(t) = \cos(2\pi f_0 t)$, consisting of two Dirac delta impulses located at $f = f_0$ and $f = -f_0$: $\frac{1}{2}\delta(f - f_0) + \frac{1}{2}\delta(f + f_0)$.

Occupied Power Bandwidth (Voice Signal)

The frequency range containing $98\%$ to $99\%$ of total signal power, which for human voice occupies up to approximately $2\,\text{kHz}$ (specifically $1675.8\,\text{Hz}$ measured for a $99\%$ region).

Bandwidth of a Digital Data Signal

The width of the main spectral lobe; for example, a $300\,\text{bps}$ BPSK signal has a main lobe bandwidth of $600\,\text{Hz}$ centered at the carrier frequency.

Circuit Bandwidth

The range of frequencies over which a circuit's output power remains greater than or equal to $-3\,\text{dB}$ (half power) relative to the passband response.

Cut-off Frequency ($f_c$)

The frequency boundary at which the power drops to half of its maximum value ($-3\,\text{dB}$ point), where the capacitive reactance equals resistance ($R_1 = X_{C1}$).

Common-Emitter Voltage Gain (High Frequency)

The gain of a common-emitter amplifier when emitter resistor $R_E$ is bypassed/shorted at high frequencies, expressed as $-\frac{R_L}{r_e'}$, where $r_e' = 5.5\,\Omega$.

Attenuation ($A$)

The reduction in signal level across a network, defined as $A = \frac{v_{out}}{v_{in}}$; in a resistive voltage divider, it is given by $\frac{R_2}{R_1 + R_2}$.

Power Gain in Decibels ($\text{dB}$)

The logarithmic ratio of output power to input power, defined by the relationship $10 \log_{10}\left(\frac{P_{out}}{P_{in}}\right)$.

Voltage Gain in Decibels ($\text{dB}$)

The logarithmic ratio of output voltage to input voltage, calculated using the formula $20 \log_{10}\left(\frac{v_{out}}{v_{in}}\right)$.

First-Order RC Low-Pass Filter

A passive filter with a series resistor $R_1$ and shunt capacitor $C_1$ that allows lower frequencies to pass and attenuates higher frequencies at a rate of $-20\,\text{dB/decade}$ beyond $f_c = \frac{1}{2\pi R_1 C_1}$.

First-Order RC High-Pass Filter

A passive filter composed of a series capacitor $C_1$ and shunt resistor $R_1$ that passes frequencies above the cutoff frequency $f_c = \frac{1}{2\pi R_1 C_1}$ while blocking DC and low frequencies.

Band-Pass Filter

A frequency-selective circuit constructed by cascading a low-pass filter and a high-pass filter to extract a desired transmission channel while blocking frequencies outside that band.

Filter Order

The degree of a filter that determines its roll-off slope; a first-order filter has a slope of $20\,\text{dB/decade}$, a second-order filter has $40\,\text{dB/decade}$, and a third-order filter has $60\,\text{dB/decade}$.

Second-Order Active Low-Pass Butterworth Filter

An active operational amplifier circuit that achieves a $-40\,\text{dB/decade}$ roll-off with cutoff frequency $f_H = \frac{1}{2\pi \sqrt{R_2 R_3 C_2 C_3}}$ and passband gain $A_F = 1 + \frac{R_F}{R_1}$.

Thermal Noise

Circuit noise generated by the random motion of electrons within conductors and resistors, described by noise power $P_n = kTB$, where $T$ is temperature and $B$ is bandwidth.

Shot Noise

A type of internal circuit-generated noise that occurs due to discrete charge carriers crossing semiconductor junctions.

Noise Factor ($F$) and Noise Figure (NF)

Measures of signal-to-noise ratio degradation through a network: Noise Factor is $F = \frac{SNR_{in}}{SNR_{out}} = \frac{G N_I + N_A}{G N_I}$, and Noise Figure is $NF = 10 \log_{10}(F)$ in $\text{dB}$.

Receiver Sensitivity ($P_{sen}$)

The minimum input signal power required by a receiver to achieve acceptable performance, defined as $P_{sen} = \text{Noise floor} + NF + SNR_{needed}$.