FM Modulator Baseband
Modulate using FM method
 Library:
Communications Toolbox / Modulation / Analog Baseband Modulation
Description
The FM Modulator Baseband block applies frequency modulation to a real input signal and returns a complex output signal.
Ports
Input
Output
Parameters
Model Examples
Block Characteristics
Data Types 

Multidimensional Signals 

VariableSize Signals 

Algorithms
A frequencymodulated passband signal, Y(t), is given as
$$Y(t)=A\mathrm{cos}\left(2\pi {f}_{\text{c}}t+2\pi {f}_{\text{\Delta}}{\displaystyle {\int}_{0}^{t}x(\text{\tau})d\text{\tau}}\right)\text{\hspace{0.17em}},$$
where:
A is the carrier amplitude.
f_{c} is the carrier frequency.
x(τ) is the baseband input signal.
f_{Δ} is the frequency deviation in Hz.
The frequency deviation is the maximum shift from f_{c} in one direction, assuming x(τ) ≤ 1.
A baseband FM signal can be derived from the passband representation by downconverting the passband signal by f_{c} such that
$$\begin{array}{c}{y}_{\text{s}}(t)=Y(t){e}^{j2\pi {f}_{\text{c}}t}=\frac{A}{2}\left[{e}^{j\left(2\text{\pi}{f}_{\text{c}}t+2\text{\pi}{f}_{\text{\Delta}}{\displaystyle {\int}_{0}^{t}x(\text{\tau})d\text{\tau}}\right)}+{e}^{j\left(2\text{\pi}{f}_{\text{c}}t+2\text{\pi}{f}_{\text{\Delta}}{\displaystyle {\int}_{0}^{t}x(\text{\tau})d\text{\tau}}\right)}\right]{e}^{j2\text{\pi}{f}_{\text{c}}t}\\ =\frac{A}{2}\left[{e}^{j2\text{\pi}{f}_{\text{\Delta}}{\displaystyle {\int}_{0}^{t}x(\text{\tau})d\text{\tau}}}+{e}^{j4\text{\pi}{f}_{\text{c}}tj2\text{\pi}{f}_{\text{\Delta}}{\displaystyle {\int}_{0}^{t}x(\text{\tau})d\text{\tau}}}\right]\text{\hspace{0.17em}}.\end{array}$$
Removing the component at 2f_{c} from y_{S}(t) leaves the baseband signal representation, y(t), which is given as
$$y(t)=\frac{A}{2}{e}^{j2\pi {f}_{\Delta}{\displaystyle {\int}_{0}^{t}x(\tau )d\tau}}.$$
The expression for y(t) can be rewritten as $$y(t)=\frac{A}{2}{e}^{j\varphi (t)}$$, where $$\varphi (t)=2\text{\pi}{f}_{\Delta}{\displaystyle {\int}_{0}^{t}x(\tau )d\tau}$$. Expressing y(t) this way implies that the input signal is a scaled version of the derivative of the phase, ϕ(t).
To recover the input signal from y(t), use a baseband delay demodulator, as this figure shows.
Subtracting a delayed and conjugated copy of the received signal from the signal itself results in this equation.
$$w(t)=\frac{{A}^{2}}{4}{e}^{j\varphi (t)}{e}^{j\varphi (tT)}=\frac{{A}^{2}}{4}{e}^{j\left[\varphi (t)\varphi (tT)\right]}\text{\hspace{0.17em}},$$
where T is the sample period. In discrete terms,
$$\begin{array}{l}{w}_{\text{n}}=w(\text{n}T),\\ {w}_{\text{n}}=\frac{{A}^{2}}{4}{e}^{j\left[{\varphi}_{\text{n}}{\varphi}_{\text{n}1}\right]}\text{\hspace{0.17em}}\text{,and}\\ {v}_{\text{n}}={\varphi}_{\text{n}}{\varphi}_{\text{n}1}\text{\hspace{0.17em}}.\end{array}$$
The signal v_{n} is the approximate derivative of ϕ_{n} such that v_{n} ≈ x_{n}.
References
[1] Hatai, I., and I. Chakrabarti. “A New HighPerformance Digital FM Modulator and Demodulator for SoftwareDefined Radio and Its FPGA Implementation.” International Journal of Reconfigurable Computing (December 25, 2011): 110. https://doi.org/10.1155/2011/342532.
[2] Taub, H., and D. Schilling. Principles of Communication Systems. McGrawHill Series in Electrical Engineering. New York: McGrawHill, 1971, pp. 142–155..