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.