Frequency Sampling Method for FIR Filters - Visual Guide

Let's explore how we design digital filters by understanding their frequency response.We start with a coordinate system where the x-axis represents frequency in radians per sample, and the y-axis shows the filter's magnitude response.For a lowpass filter, we want to pass low frequencies and block high frequencies. The ideal response looks like this.We can divide the frequency response into three main regions. First, the passband, where signals pass through unchanged.Next, the transition band, where the response gradually decreases from pass to stop.Finally, the stopband, where high-frequency signals are blocked or heavily attenuated.To design a digital filter, we need to sample this continuous frequency response at N equally spaced points.Here we're using sixteen points to sample our desired frequency response. Each point represents how the filter should behave at that specific frequency.These sample points will be used to calculate the actual filter coefficients through an inverse Fourier transform.To transform our frequency response into filter coefficients, we need to use the Inverse Discrete Fourier Transform.We start with N equally spaced frequency samples. These points represent our desired frequency response.For real-valued filter coefficients, these frequency samples must exhibit conjugate symmetry.The Inverse Discrete Fourier Transform converts these frequency samples into time-domain coefficients using this equation.Through this transformation, our N frequency samples become N time-domain filter coefficients.Each frequency sample contributes to every time-domain coefficient through this complex mathematical relationship.The phase response must also meet specific symmetry requirements to ensure real-valued coefficients.Now that we have our filter coefficients, let's examine the practical aspects of implementation.Here's our ideal frequency response - a perfect lowpass filter with a cutoff at π.However, when we truncate the filter to a finite length, we get ripples in both the passband and stopband. This is known as the Gibbs phenomenon.To understand how we can improve this, let's look at the Hamming window function.The window function gradually tapers the filter coefficients, reducing the abrupt truncation that causes these ripples.Let's examine the key characteristics of our filter design.There are important tradeoffs to consider in filter design. A longer filter gives better frequency response but increases computational complexity.Comparing the three responses - ideal, truncated, and windowed - we can see how windowing helps achieve a practical compromise between the ideal and truncated responses.