Analog modulation formulas

Carrier plus two sidebands, each of amplitude $\mu A_c/2$. Bandwidth $B=2f_m$ (for a general message, $B=2W$ where $W$ is the highest message frequency).

These power relations hold for any $\mu$. Without overmodulation ($\mu \le 1$) the efficiency is at most $1/3$, reached at $\mu =1$. Each sideband carries ${\mu }^2{P}_c/4$. In terms of antenna (load) current, $I_t=I_c\sqrt{1+{\mu }^2/2}$.

For a message made of tones of different frequencies, each with its own index ${\mu }_i$. Use ${\mu }_t$ in the power and efficiency formulas. The condition ${\mu }_1+{\mu }_2+\cdots \le 1$ is enough to keep the envelope from going negative.

$A_{\max }$ and $A_{\min }$ are the maximum and minimum of the envelope, for a single tone with $\mu \le 1$. For $\mu >1$ (overmodulation) the envelope is distorted and an envelope detector fails.

The first condition (with $f_m$ replaced by $W$ for a general message) lets the capacitor smooth out the carrier ripple yet follow the envelope. The second is the condition, for a single tone of index $\mu <1$, that avoids diagonal clipping (the capacitor discharging slower than the envelope falls).

Power in a $1\Omega$ load, for a zero-mean message. The spectrum has no carrier component. The envelope is $A_c|m(t)|$, not $m(t)$, so an envelope detector cannot be used; DSB-SC needs coherent detection.

DSB-SC input, local carrier of unit amplitude $\cos ({\omega }_{c}t+\varphi )$ or $\cos ({\omega }_c+\Delta \omega )t$, followed by a low-pass filter. The output vanishes at $\varphi =90^{\circ }$ (quadrature null effect), and a frequency error gives a beating (fading) output. For SSB, a phase error causes phase distortion of the output, and a frequency error shifts every output tone by $\Delta f$. A Costas loop or a squaring loop recovers a synchronous carrier for DSB-SC.

Upper sign gives USB, lower sign gives LSB. $\hat{m}(t)$ is the Hilbert transform of $m(t)$ and has the same power as $m(t)$. Power in a $1\Omega$ load. Coherent detection with a unit-amplitude local carrier gives $\frac{A_c}{4}m(t)$.

$k_p$ in rad/V, $k_f$ in Hz/V. Instantaneous frequency: FM $f_i=f_c+k_{f}m(t)$; PM $f_i=f_c+\frac{k_p}{2\pi }\frac{dm}{dt}$. FM by $m(t)$ is PM by $\int mdt$, and PM by $m(t)$ is FM by $dm/dt$.

In FM, $\Delta f$ depends only on the amplitude of the message and $\beta$ falls as $f_m$ rises. In PM, $\beta$ depends only on the amplitude and $\Delta f$ grows with $f_m$.

Spectral lines at $f_c\pm n{f}_m$ with amplitudes $A_c|J_n(\beta )|$. The carrier amplitude $A_c{J}_0(\beta )$ vanishes at $\beta \approx 2.405,5.520,8.654,\dots$

In a $1\Omega$ load. Total power is independent of $\beta$; modulation only redistributes power from the carrier to the sidebands. Power in the carrier plus $k$ pairs of sidebands is $\frac{A_c^2}{2}[J_0^2+2{\sum }_{n=1}^k{J}_n^2]$.

An approximation; for a single tone it contains roughly 98 percent of the total power. For $\beta \ll 1$, $B\approx 2f_m$ (narrowband); for $\beta \gg 1$, $B\approx 2\Delta f$ (wideband). For a general message use $B=2(\Delta f+W)=2W(D+1)$, with peak deviation $\Delta f=k_f\max |m|$ and deviation ratio $D=\Delta f/W$.

Coherent detection for DSB-SC and SSB; envelope detection for AM above threshold, with $\mu \le 1$, so its figure of merit is at most $1/3$; FM above threshold without pre-emphasis. For a general zero-mean message of power $P$: AM gives $k_a^{2}P/(1+k_a^{2}P)$ and FM gives $3k_f^{2}P/W^2$. FM beats fully modulated ($\mu =1$) AM when $\frac{3}{2}{\beta }^2>\frac{1}{3}$, that is $\beta >\sqrt{2}/3\approx 0.47$. Above threshold, FM trades bandwidth for noise performance.

The noise at an FM discriminator's output has a PSD proportional to $f^2$, so high-frequency message components suffer most. The transmitter boosts the high frequencies (pre-emphasis) and the receiver cuts them by the same amount (de-emphasis). The overall response to the message is flat, while the high-frequency noise is reduced. A $75\mu \text{s}$ time constant corresponds to $f_1\approx 2.12$ kHz.

$f_s$ is the wanted station frequency. The image station lies at $f_{LO}+f_{IF}$ and also mixes down to $f_{IF}$, causing interference. With a low-side LO ($f_{LO}=f_s-f_{IF}$), $f_{im}=f_s-2f_{IF}$.

For a single tuned circuit of loaded quality factor $Q$ at the RF stage. $\alpha$ is a voltage ratio; in dB it is $20{\log }_{10}\alpha$. With identical tuned stages before the mixer, their ratios multiply. A larger IF moves the image farther from $f_s$ and improves image rejection; a smaller IF makes adjacent-channel selectivity easier to achieve.

$B_{guard}$ is the guard band between adjacent channels. For $N$ identical SSB channels of bandwidth $W$, $B_{total}=NW+(N-1)B_{guard}$. For DSB-SC or AM channels each $B_k=2W$.

$N$ channels, each band-limited to $W$ and sampled at $f_{\text{samp}}$ (the frame rate), at or above the Nyquist rate $2W$. The minimum (Nyquist) transmission bandwidth is half the pulse rate: $N{f}_{\text{samp}}/2$ for PAM-TDM and $R_b/2$ for binary PCM-TDM. Add synchronisation bits per frame if given.