This chapter unifies filter analysis and design concepts into a practical pathway for building realizable FIR filters. Starting from the duality between time-domain convolution and frequency-domain multiplication, we connect impulse response, frequency response, and the z-transform. We motivate the ideal (“brick-wall”) low-pass filter and then show why its sinc impulse response is noncausal and infinite. To obtain realizable FIR designs, we introduce the window method: truncating and shaping the ideal impulse response to control transition width and stopband attenuation. We examine standard windows (rectangular, Hann, sine, Kaiser, Vorbis), discuss the Gibbs phenomenon, and present an anti-aliasing example relevant to downsampling. Finally, we illustrate optimization-based window/filter design and modulation to obtain high-pass and band-pass filters. Throughout, Python code supports analysis and visualization.