# Complex filters used in analog domain [closed]

Complex filters can be used in low IF architecture especially in DSP in order to phase out (reject) the image signal. How are complex filters implemented in the analog domain?

• what does "phase out" mean? And analog filters depend really on the kind of signals, their amplitude and frequency, and the quality of filtering you need. Generally, you'd not implement similar filters in digital as in analog domain, so you're inherently asking to compare incomparable things. Please narrow down your question by stating the technological field and at least the IF you're referring to! Otherwise, your question is too broad. – Marcus Müller Oct 5 '18 at 15:43
• also, please state what "efficient" means to you. Efficient in power? Efficient in cost? Efficient in space used? Efficient in design effort? Efficient in weight? Efficient in "making my boss who's super into analog filters happy"? – Marcus Müller Oct 5 '18 at 15:46
• Thank you very much Dan for taking the time to understand and answer all my questions even though they may sometimes be "too broad". – Hatem Tawfik Oct 5 '18 at 16:51
• So if we compare the analog implementation of the diagram you are showing with the digital one, in terms of image cancellation, what would be the issues there? – Hatem Tawfik Oct 5 '18 at 16:55
• I think as you asked it, it was too broad. I updated to be more specific to your question. The issues with image cancellation and DC offset are amplitude balance and phase balance which will typically be worst in the analog world. You can see this by realizing that a single sideband tone given by $Ae^{j\omega t}$ is represented using Euler's identity as $Acos(\omega t)+ jAsin(\omega t)$ Here we have two real signals in perfect quadrature and perfect amplitude balance. Resulting in one single sideband. Try it with an ampltude and phase imbalance and you will see how the sideband is created. – Dan Boschen Oct 5 '18 at 18:14

Vector Modulators and Single Sideband Mixer (frequency translator) are the common and relatively simple analog component for rejecting the image signal.

The top block diagram in this post is the architecture of a vector modulator and single sideband mixer and explains in further detail how it provides image rejection using complex signalling.

Frequency shifting of a quadrature mixed signal

Normally we see these components with a real input where an internal 90° splitter immediately converts the input to a complex signal and then it is further processed. However there is no reason these can't also be used in identical fashion with the quadrature splitter bypassed in the case of a low IF IQ signal already exists.

This would be the favored approach for eliminating the image signal.

However to implement a generic complex filter in the analog domain, you would need to do the following with four real filters as shown in the diagram below. This is related to performing full complex multiplication with real signals as given by:

$$(s_I+js_Q)(c_I-jc_Q)$$ $$= (s_Ic_I+s_Qc_Q) + j(s_Ic_Q-s_Qc_I)$$

Corresponding to the multiplication of a signal (s) with a coefficient such as in an FIR filter (c) showing that we have an I and Q input, multiplied by and I an Q coefficient, resulting in a I (real part given by $$s_Ic_I+s_Qc_Q$$) and Q (Imaginary part given by s_Ic_Q-s_Qc_I).

Equivalently complex analog filtering (the filtering of a complex input with a complex impulse response) is formed with four real filters as shown in the graphic below. Each filter has a real impulse response given by $$h_i(t)$$ or $$h_q(t)$$ corresponding the real and imaginary portion of the complex impulse response desired. Also to add, I did come across this: https://ieeexplore.ieee.org/abstract/document/4151912

Note that SAW filters are an analog equivalent of an FIR filter ("all=zero" filter, with poles at infinity, formed by the summation of weighted delays)

• to add a bit on SAW filters: As these require relatively specific materials and dimensions, you can typically only easily get SAW filters for specific bands – which thus happen to be "typical" IF bands. The fact that for certain non-zero IFs there's good analog filters is the reason why in high-sensitivity applications, you often do not do direct zero-IF conversion but have a superhet (or hybrid) receiver architecture. – Marcus Müller Oct 5 '18 at 18:21
• Good point Marcus! I often started a design / frequency plan based on what filters were available. Volume drives the availability and technology drives the upper and lower limits but 10's of MHz to 100's of MHz is typical. FBAR technology for microwave (cell band) frequencies with impressive results as long as high power is not required. – Dan Boschen Oct 5 '18 at 19:01