-rep

Discrete-time Fourier Transform (DTFT)
• Consider the 2-pt moving average
𝐻 𝑧 = 1
2 + 1
2 𝑧−1 𝑧 > 0 𝐻 𝑤 = 1
2 + 1
2 𝑒−𝑗𝑤 = 1
2 + 1
2 𝑐𝑜𝑠 𝑤 − 𝑗 1
2 𝑠𝑖𝑛 𝑤
𝐻 𝑤 = 1
2 + 1
2 𝑐𝑜𝑠 𝑤
𝜃 𝑤 = −𝑎𝑡𝑎𝑛 𝑠𝑖𝑛 𝑤
1 + 𝑐𝑜𝑠 𝑤

Discrete-time Fourier Transform (DTFT)
• Consider a causal bandpass system with the following pole-zero
placement
𝐻 𝑧 = 1 − 𝑟2
2
1 − 𝑧−2
1 − 2𝑟𝑐𝑜𝑠 𝑤0 𝑧−1 + 𝑟2𝑧−2 𝐻 𝑤 = 1 − 𝑟2
2
1 − 𝑧−2
1 − 2𝑟𝑐𝑜𝑠 𝑤0 𝑒−𝑗𝑤 + 𝑟2𝑒−𝑗2𝑤
𝑟 = 0.75
𝑎 = 0.95
𝑤𝑜 = 0.5𝜋

Discrete-time Fourier Transform (DTFT)
• Consider an FIR system with the following transfer function
𝐻 𝑧 = 1
2 − 1
2𝑧−5 𝐻 𝑤 = 1
2 − 1
2𝑐𝑜𝑠 5𝑤 + 𝑗1
2𝑠𝑖𝑛 5𝑤

Discrete-time Fourier Transform (DTFT)
• Consider an exponential sequence (−1 < 𝑎 < 0,causal, and stable)
1 + 𝑎 𝑎𝑛𝑢 𝑛 ↔ 1 + 𝑎
1 − 𝑎𝑧−1 𝑅𝑂𝐶 𝑧 > 𝑎 𝐻 𝑤 = 1 + 𝑎
1 − 𝑎𝑒−𝑗𝑤 = 1 + 𝑎
1 − 𝑎𝑐𝑜 𝑠 𝑤 + 𝑗𝑎𝑠𝑖𝑛 𝑤
𝜃 𝑤 = −𝑎𝑡𝑎𝑛 𝑎𝑠𝑖𝑛 𝑤
1 − 𝑎𝑐𝑜𝑠 𝑤
𝐻 𝑤 = 1 + 𝑎
1 + 𝑎2 − 2𝑎𝑐𝑜𝑠 𝑤
𝑎 = −1
2
𝑎 = −3
4
Highpass

Discrete-time Fourier Transform (DTFT)
• Consider an exponential sequence (0 < 𝑎 < 1,causal, and stable)
1 − 𝑎 𝑎𝑛𝑢 𝑛 ↔ 1 − 𝑎
1 − 𝑎𝑧−1 𝑅𝑂𝐶 𝑧 > 𝑎 𝐻 𝑤 = 1 − 𝑎
1 − 𝑎𝑒−𝑗𝑤 = 1 − 𝑎
1 − 𝑎𝑐𝑜 𝑠 𝑤 + 𝑗𝑎𝑠𝑖𝑛 𝑤
𝐻 𝑤 = 1 − 𝑎
1 + 𝑎2 − 2𝑎𝑐𝑜𝑠 𝑤
𝜃 𝑤 = −𝑎𝑡𝑎𝑛 𝑎𝑠𝑖𝑛 𝑤
1 − 𝑎𝑐𝑜𝑠 𝑤
𝑎 = 1
2
𝑎 = 3
4
lowpass