Digital Signal Processing: FIR Filter Design

Digital Signal Processing Lecture Notes

• Lecture is on FIR Filter Design for a 4th-Year Communication Department
• Lecturer is Dr. Ibrahim el metwally from Misr Engineering and Technology (MET)

FIR Filter Design Theorem

• A necessary and sufficient condition for linear phase in a discrete-time system with impulse response h(n) is that h(n) has finite duration N and is symmetric about its midpoint.
• h(n) = h_a(n), for N₁ ≤ n ≤ N₂
• h(n) = 0, otherwise

Frequency Response

• Figure 4.13 shows the frequency response obtained by rectangularly windowing an ideal low-pass impulse response.
• The frequency response is represented as H(e^jw) which is equal to h(n) * w(n)

Window Equations

• Rectangular: w_R(n) = 1, 0 ≤ n ≤ N-1; 0, elsewhere
• Bartlett: w_B(n) = 2n/(N-1), 0 ≤ n ≤ (N-1)/2; 2 - 2n/(N-1), (N-1)/2 ≤ n ≤ N-1; 0, elsewhere
• Hanning: w_Han(n) = [1 - cos(2πn/(N-1))]/2, 0 ≤ n ≤ N-1; 0, elsewhere
• Hamming: w_Ham(n) = 0.54 - 0.46 cos(2πn/(N-1)), 0 ≤ n ≤ N-1; 0, elsewhere
• Blackman: w_Black(n) = 0.42 - 0.5 cos(2πn/(N-1)) + 0.08 cos(4πn/(N-1)), 0 ≤ n ≤ N-1; 0, elsewhere

Design Procedure

• Step 1: Select window type based on given attenuation (K2)
• Step 2: Determine the number of points (N) in the window using N ≥ k * 2π/(ω₂ - ω₁) (refer to Table 4-3)
• Step 3: Calculate ω and α: ω = ω_c; α = (N - 1)/2
• Step 4: Calculate h(n): h(n) = [sin(ω(n - α))]/[π(n - α)] * w(n)
• Step 5: Find the frequency response

Design Table for FIR Low-Pass Filter

• Table 4.3 provides transition width and minimum stopband attenuation for different window types(Rectangular, Bartlett, Hanning, Hamming, Blackman, Kaiser)

Example Design

• Design a low-pass digital filter for an A/D-H(z)-D/A structure with a 3-dB cutoff of 30π rad/sec and 50 dB attenuation at 45π rad/sec. The sampling rate is 100 samples/sec.
• A Hamming window is chosen for its narrow transition band.

Example Calculation (Continued)

• Calculate approximate number of points (N) needed to satisfy the transition band requirement.
• The next odd integer is selected for N (e.g., 55).
• Determine ω_c and α using relevant equations.
• Calculate a trial impulse response function h(n) using the chosen window type.

Test your understanding of FIR filter design concepts as covered in the Digital Signal Processing lecture by Dr. Ibrahim el Metwally. This quiz focuses on key topics such as linear phase conditions, frequency response, and various window equations used in FIR filter implementation.