[訊號與系統] FIR系統的弦波響應 (1)

考慮 FIR 系統為 線性非時變(Linear Time-Invariant, LTI) 系統，且假設輸入為 離散 complex exponential 則其對應的輸出將非常容易計算：考慮 FIR 系統
\[
y[n] = \sum_{k=0}^M b_k x[n-k]
\]

假設輸入為 complex exponential 表為 $x[n] = x(nT_s) = A e^{j \varphi} e^{j \omega Ts n}$ 且 $-\infty < n < \infty$。則輸出為
\[\begin{array}{l}
y[n] = \sum\limits_{k = 0}^M {{b_k}} x[n - k]\\
 = \sum\limits_{k = 0}^M {{b_k}} A{e^{j\varphi }}{e^{j\omega Ts\left( {n - k} \right)}}\\
 = \left( {\sum\limits_{k = 0}^M {{b_k}} {e^{j\omega Ts\left( { - k} \right)}}} \right)A{e^{j\varphi }}{e^{j\omega Ts\left( n \right)}}\\
 := H\left( {{e^{ - j\widehat \omega }}} \right)\underbrace {A{e^{j\varphi }}{e^{j\omega Ts\left( n \right)}}}_{ = x\left[ n \right]}
\end{array}\]其中 $\widehat{\omega} := \omega T_s$ 且
\[H\left( {{e^{ - j\widehat \omega }}} \right) = \sum\limits_{k = 0}^M {{b_k}} {e^{ - j\widehat \omega k}} = \sum\limits_{k = 0}^M {h\left[ k \right]} {e^{ - j\widehat \omega k}}\]稱作 frequency-response function，一般而言我們簡稱為 frequency response。

回憶由於 FIR系統的脈衝響應 與 濾波器的係數相同，亦即 $b_k = h[k]$，我們可以將上述頻率響應改寫成
\[H\left( {{e^{ - j\widehat \omega k}}} \right) = \sum\limits_{k = 0}^M {{b_k}} {e^{ - j\widehat \omega k}} = \sum\limits_{k = 0}^M {h\left[ k \right]} {e^{ - j\widehat \omega k}}\]

Comments:
1. 輸入為離散時間訊號 $x[n] = x(nT_s) = A e^{j \varphi} e^{j \omega Ts n}$ 則 LTI FIR 系統亦為離散時間 complex exponential 乘上不同的複數大小，但頻率 $\widehat{\omega}$ 不變。

2. 上述以離散時間訊號為 complex exponential $x[n] = x(nT_s) = A e^{j \varphi} e^{j \omega Ts n}$ 為輸入，則 LTI FIR 系統可表為
\[y[n] = H\left( {{e^{ - j\widehat \omega k}}} \right)x\left[ n \right]\]

3. 注意到 $H\left( {{e^{ - j\widehat \omega k}}} \right)$ 為複數值函數，故我們可使用極座標表示法將其表示為
\[y[n] = \left( {\left| {H\left( {{e^{ - j\widehat \omega k}}} \right)} \right|{e^{j\angle H\left( {{e^{ - j\widehat \omega k}}} \right)}}} \right)A{e^{j\varphi }}{e^{j\omega Ts\left( n \right)}}\]
亦即，
\[\begin{array}{l}
y[n] = H\left( {{e^{ - j\widehat \omega k}}} \right)\underbrace {A{e^{j\varphi }}{e^{j\omega Ts\left( n \right)}}}_{ = x\left[ n \right]}\\
 = \left( {\left| {H\left( {{e^{ - j\widehat \omega k}}} \right)} \right|{e^{j\angle H\left( {{e^{ - j\widehat \omega k}}} \right)}}} \right)A{e^{j\varphi }}{e^{j\widehat \omega n}}\\
 = \left( {\left| {H\left( {{e^{ - j\widehat \omega k}}} \right)} \right|A} \right){e^{j\left( {\angle H\left( {{e^{ - j\widehat \omega k}}} \right) + \varphi } \right)}}{e^{j\widehat \omega n}}
\end{array}\]一般而言，我們會稱 ${\left| {H\left( {{e^{ - j\widehat \omega k}}} \right)} \right|}$ 為系統增益 (gain) 且 ${\angle H\left( {{e^{ - j\widehat \omega k}}} \right)}$ 稱為系統相位 (phase)。