# The desired frequency-response specifications

The complex function $D (e^{-jw})$ is defined on the domain of approximation $\Omega$ .In most cases the domain $\Omega$ is the union of several disjoint frequency bands which are separated by transition bands where no desired response is specified .We denote the union of all passbands by $\Omega^p$ and stopbands by $\Omega^s$. If the designed filter is to have real-valued coeffcients only the domain $\Omega\cap[0 ,\pi]$ is considered,what is the type of coefficients which used in the domain $[0 ,2\pi]$ or$[-\pi,\pi]$?

If you have to consider the whole frequency spectrum range in $[0,2\pi]$ for the design of the discrete-time filter, without assuming any type of symmetry, then you are considering the most general case of the filter and its coefficients will be complex valued. And nothing more can be said about them, unless you impose further constraints on the impulse and frequency responses of the filter.
• if the coefficients of the impulse response were real, then the associated frequency response would be Hermitian symmetric which implies that only half of the spectrum is independent and the remaining half depends on the first (indeed is its complex conjuage replica). Of course if a spectrum is Hermitian syymetric on the whole domain $[0,2\pi]$ then the coeff will be real. Yet in that case you need to consider only the first half $[0,\pi]$. – Fat32 Jul 26 '18 at 17:13