2.2. DTFT properties and tables#
2.2.1. Table of DTFT pairs#
\(\boldsymbol{x[n]}\) |
\(\boldsymbol{X(e^{j{\hat{\omega}}})}\) |
Condition |
|---|---|---|
\(\displaystyle a^n u[n]\) |
\(\displaystyle \frac{1}{1 - a e^{-j \hat{\omega}}}\) |
\(|a| < 1\) |
\(\displaystyle (n+1) a^n u[n]\) |
\(\displaystyle \frac{1}{(1-a e^{-j \hat{\omega}})^2}\) |
\(|a| < 1\) |
\(\displaystyle \frac{(n+r-1)!}{n! (r-1)!} a^n u[n]\) |
\(\displaystyle \frac{1}{(1-a e^{-j \hat{\omega}})^r}\) |
\(|a| < 1\) |
\(\displaystyle \delta[n]\) |
\(\displaystyle 1\) |
|
\(\displaystyle \delta[n - n_0]\) |
\(\displaystyle e^{-j \hat{\omega} n_0}\) |
|
\(\displaystyle u[n]-u[n-N]\) |
\(\displaystyle \frac{\sin(N\hat{\omega}/2)}{\sin(\hat{\omega} / 2)} e^{-j(\frac{N-1}{2})\hat\omega}\) |
|
\(\displaystyle \frac{\sin(\hat\omega_0 n)}{\pi n}\) |
\( X(e^{j{\hat{\omega}}}) = \begin{cases} 1, & 0 \leq |\hat{\omega}| \leq \hat\omega_0 \\ 0. & \hat\omega_0 < |\hat{\omega}| \leq \pi \end{cases}\) |
|
\(\displaystyle 1\) |
\(\displaystyle 2 \pi \delta(e^{j\hat{\omega}})\) |
|
\(\displaystyle u[n]\) |
\(\displaystyle \frac{1}{1-e^{-j \hat{\omega}}} + \pi \delta(e^{j\hat{\omega}})\) |
|
\(\displaystyle e^{j \hat{\omega}_0 n}\) |
\(\displaystyle 2 \pi \delta(e^{\hat{\omega} - \hat{\omega}_0})\) |
|
\(\displaystyle \cos(\hat{\omega}_0 n)\) |
\(\displaystyle \pi \left[ \delta(e^{j(\hat{\omega}-\hat{\omega}_0)}) + \delta(e^{j(\hat{\omega}+\hat{\omega}_0)}) \right]\) |
|
\(\displaystyle \sin(\hat{\omega}_0 n)\) |
\(\displaystyle \frac{\pi}{j} \left[ \delta(e^{j(\hat{\omega}-\hat{\omega}_0)}) - \delta(e^{j(\hat{\omega}+\hat{\omega}_0)}) \right]\) |
|
\(\displaystyle \sum_{k=-\infty}^{\infty} \delta[n - kN]\) |
\(\displaystyle \frac{2 \pi}{N} \sum_{k=0}^{N-1} \delta\left(e^{j\left(\hat{\omega} - \frac{2 \pi k}{N} \right)} \right)\) |
2.2.2. Table of DTFT properties#
For each property listed below, assume \(x[n] \stackrel{\text{DTFT}}{\longleftrightarrow} X(e^{j\hat\omega})\) and \(y[n] \stackrel{\text{DTFT}}{\longleftrightarrow} Y(e^{j\hat\omega})\):
Property |
Time domain |
Frequency domain |
|---|---|---|
Linearity |
\(\alpha x[n] + \beta y[n]\) |
\(\alpha X(e^{j{\hat{\omega}}}) + \beta Y(e^{j{\hat{\omega}}})\) |
Time Shifting |
\(x[n-n_0]\) |
\(X(e^{j{\hat{\omega}}}) e^{-j \hat{\omega} n_0}\) |
Frequency Shifting |
\(x[n] e^{j \hat{\omega}_0 n}\) |
\(X(e^{j(\hat{\omega}-\hat{\omega}_0)})\) |
Conjugation |
\(x^*[n]\) |
\(X^*(e^{-j\hat{\omega}})\) |
Time Reversal |
\(x[-n]\) |
\(X(e^{-j\hat{\omega}})\) |
Convolution |
\(x[n]*y[n]\) |
\(X(e^{j\hat{\omega}}) Y(e^{j\hat{\omega}})\) |
Multiplication |
\(x[n] y[n]\) |
\(\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\theta}) Y(e^{j(\hat{\omega}-\theta)}) d\theta\) |
Time Differencing |
\(x[n]-x[n-1]\) |
\((1-e^{-j\hat{\omega}})X(e^{j\hat{\omega}})\) |
Accumulation |
\(\displaystyle \sum_{k=-\infty}^{n} x[k]\) |
\(\displaystyle \frac{X(e^{j\hat{\omega}})}{1 - e^{-j\hat{\omega}}} + \pi X(e^{j0}) \delta(e^{j\hat{\omega}})\) |
Frequency Differentiation |
\(n x[n]\) |
\(\displaystyle j\frac{d X(e^{j\hat{\omega}})}{d\hat{\omega}}\) |
Parseval Theorem |
\(\displaystyle \sum_{n=-\infty}^{\infty} x[n] y^*[n]\) |
\(\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\hat\omega}) Y^*(e^{j\hat\omega}) d\hat\omega\) |
2.2.3. Use-of-tables example#
Consider the window signal \(w_{\lambda}[n] = \lambda^{|n|}\), where \(\lambda \in (0,1)\). Clearly, \(w_{\lambda}[n] \in \ell^1\) and hence its DTFT \(W_{\lambda}(e^{j\hat\omega})\) exists.
To use the tables above to find \(W_{\lambda}(e^{j\hat\omega})\), let \(\tilde{w}[n] = \lambda^n u[n]\) and notice that
Looking up the DTFT-pair table, \(\tilde{w}[n] \stackrel{\text{DTFT}}{\longleftrightarrow} \tilde{W}_{\lambda}(e^{j\hat\omega}) = \frac{1}{1 - \lambda e^{-j \hat{\omega}}}\).
Then using the linearity, time shifting, and time reversal properties, we get