# Understanding FSK and FHSS combination

I am having trouble to understand how to combine Frequency Shift Keying (FSK) with Frequency Hopping Spread Spectrum (FHSS). As far as I understand, A IQ FSK-modulated signal $$s(t)$$ can be written as $$s(t) = \Re \{ A\exp[j(2\pi f_c(t) t + \varphi)] \},$$ where the instantaneous frequency $$f_c(t)$$ is given by $$f_c(t) = \sum_{k \in \mathbb{Z}} A_k(t) g(t - kT_s) + f_0(t) \quad \text{with } A_k(t) \, \in \{-1, +1\} \, (\forall k, \, \forall t)$$ where $$g(\cdot)$$ is the pulse-shaping function ($$T_s$$ being the symbol duration), and $$f_0(t)$$ is the instantaneous carrier frequency, in the given hop set.

My questions are:

• are the expressions of $$s(t)$$ and $$f_c(t)$$ given above valid ?
• is my understanding of FSK and FHSS correct ?
• Your $f_c(t)$ is a complex-valued function, and so when multiplied by $j2\pi$ etc gives a real-valued function in the argument of the $\exp$. Thus, $\exp[j(2\pi f_c(t)t+\cdots] = \exp[r(t) + j\cdots]$ so that $s(t)$ is actually a sinusoid with exponentially increasing or decreasing amplitude, and not FSK at all. Also, (slow) FH systems generally transmit multiple bits at the same frequency-hopped carrier frequency before the carrier is hopped to the next frequency in the frequency-hopping pattern. Mar 16, 2022 at 20:55
• Hello @DilipSarwate, thank you for your answer. You are right, if I have the term $B_k(t)$ in the expression of $f_c(t)$ I will indeed have a real component inside the exponential, thanks for pointing that out. So, if I only keep the $A_k(t)$, is the expression then that of a FSK signal ? Have a nice day, Alex. Mar 17, 2022 at 8:54
• Please edit your question to remove $B_k$ etc, and while you are at it, delete the last sentence. Thanks and pleasantries such as “Have a nice day” are specifically called out as undesirable. You can edit your question by clicking on the link marked Edit below your post. Mar 17, 2022 at 13:32