# Real and complex low pass equivalent of band-pass signal

A real band-pass signal $s(t)$ with center frequency $f_c$ can always be represented as

$$s(t) = A(t) \cos(2\pi f_c t)$$

where $A(t)$ is real and can be viewed as amplitude.

Also

$$s(t) = A(t) \cos(2\pi f_ct) = \Re\left\{(A(t) e^{j2\pi f_ct})\right\}$$

Then we consider

$$s_l(t)= A(t) e^{j2\pi f_ct}$$

to be the corresponding complex band-pass signal of $s(t)$.

My question is how can I find the real low pass equivalent of $s(t)$ and the complex low pass equivalent of $s_l(t)$?

• it's $A(t)$. for both (because $A(t)$ must be real because $s(t)$ is, then the Fourier transform of $A(t)$ (geez, what letter do i use if cap $A$ is the time-domain function?) is Hermitian Symmetric (i think that was the term). if you want to model $A(t)$ (and i would call it "$a(t)$" if i were doing this) as having an asymmetric spectrum, then i would make it complex for $s_l(t)$. – robert bristow-johnson Jul 25 '15 at 17:28
• @robertbristow-johnson thanks, I would like to know one more thing though. Is there a process that I can follow to obtain the complex band pass signal from the real band pass signal? – Dimitri C Jul 25 '15 at 17:39
• yes. it's called the Analytic signal. you get it by adding your original real signal $s(t)$ with the Hilbert transform of it (times $j$ ). – robert bristow-johnson Jul 25 '15 at 17:45
• @robertbristow-johnson Thanks again Robert. One last thing. Can any of the signals I mentioned above be characterized as low pass signals or as high pass signals? – Dimitri C Jul 25 '15 at 18:09

It's not true that any real-valued band pass signal can be written as shown in your first equation:

$$s(t)=A(t)\cos(\omega_c t)\tag{1}$$

The most general representation of a real-valued band pass signal is given by

$$s(t)=\Re\{a(t)e^{j\omega_c t}\}\tag{2}$$

where $a(t)$ is a complex-valued function, called the complex envelop of $s(t)$. It is the complex baseband (low pass) representation of $s(t)$. With $a(t)=a_I(t)+ja_Q(t)$, (2) can be written as

$$s(t)=a_I(t)\cos(\omega_c t)-a_Q(t)\sin(\omega_c t)\tag{3}$$

where $a_I(t)$ and $a_Q(t)$ are called the in-phase and the quadrature components, respectively. Both $a_I(t)$ and $a_Q(t)$ are real-valued baseband (low pass) signals.

A third, equivalent, representation of $s(t)$ is obtained by writing the complex envelop as

$$a(t)=b(t)e^{j\phi(t)}\tag{4}$$

where $b(t)$ is the (real-valued) envelop, and $\phi(t)$ is the phase. Using (4) and (2), the band pass signal $s(t)$ can be written as

$$s(t)=b(t)\cos(\omega_ct+\phi(t))\tag{5}$$

Equations (2), (3) and (5) are three equivalent representations of a general real-valued band pass signal. If you compare representation (5) to Equation (1), you see that Equation (1) is a special case for $\phi(t)=2\pi k$, $k=0,1,\ldots$, i.e. for a constant phase which is zero (or a multiple of $2\pi$). From Equation (3), this is equivalent with the condition that the quadrature component $a_Q(t)$ is zero.

Note that the complex-valued band pass signal

$$\tilde{s}(t)=a(t)e^{j\omega_c t}\tag{6}$$

is the analytic signal mentioned in a comment. It has no negative frequency components and it can be obtained from $s(t)$ by computing

$$\tilde{s}(t)=s(t)+j\mathcal{H}\{s(t)\}\tag{7}$$

where $\mathcal{H}\{\cdot\}$ denotes the Hilbert transform.