4.2. Short-time DTFT#

  • Many practical signals, such as musical signals, have time-varying spectrums. That is, their frequency contents (i.e., FT or DTFT) change over time.

  • For example, consider a discrete-time signal \(x[n]\) that is obtained by oversampling a piece of music. The DTFT \(X(e^{j\hat\omega})\) of the whole \(x[n]\) contains all the frequency contents (i.e., all the notes played) over the whole piece of music. In particular, we can’t tell from \(X(e^{j\hat\omega})\) which note is played at which time. This means that the time resolution of \(X(e^{j\hat\omega})\) is poor. The uncertainty principle thus implies that we may be able to obtain a fine frequency resolution.

  • We may employ the windowing approach to improve time resolution:

    • Consider a discrete-time window signal \(\tilde{w}[n]\) whose energy is concentrated around \(n=0\) and have a small \(\sigma_t\) (in the interpretation for discrete-time signals given in Section 4.1).

    • Apply windowing to the time of interest, say \(n=m\), by multiplying \(x[n]\) with \(\tilde{w}[n-m]\).

    • Take DTFT of the windowed signal \(x[n] \tilde{w}[n-m]\).

    Intuitively, windowing \(x[n]\) by \(\tilde{w}[n-m]\) focuses our attention to the parts of \(x[n]\) around the time \(m\).

  • Note that different DTFTs may be obtained at different times of interest. The collection of all such DTFTs over the range of \(m\) is called the short-time DTFT \(X(e^{j\hat\omega}; m)\) of \(x[n]\):

    \[\begin{equation*} X(e^{j\hat\omega}; m) = \sum_{n=-\infty}^{\infty} x[n] \tilde{w}[n-m] e^{-j\hat\omega n}. \end{equation*}\]

  • From the uncertainty principle, we know that for each fixed \(m\), we will suffer from a poorer frequency resolution by using \(X(e^{j\hat\omega}; m)\) instead of \(X(e^{j\hat\omega})\) because the time resolution of the windowed signal \(x[n] \tilde{w}[n-m]\) is finer than that of the whole signal \(x[n]\).

  • To further see this tradeoff, let \(\tilde{w}[n] \stackrel{\text{DTFT}}{\longleftrightarrow} \tilde{W}(e^{j\hat\omega})\). Since \(X(e^{j\hat\omega}; m)\) is the DTFT of \(x[n] \tilde{w}[n-m]\), the time-shifting and multiplication properties of DTFT gives

    \[\begin{equation*} X(e^{j\hat\omega}; m) = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\theta}) \tilde{W}(e^{j(\hat\omega -\theta)}) e^{j(\hat\omega -\theta)m} \,d\theta. \end{equation*}\]

    That is, \(X(e^{j\hat\omega}; m)\) is the circular convolution between \(X(e^{j\hat\omega})\) and \(\tilde{W}(e^{j\hat\omega}) e^{-j\hat\omega m}\) in the frequency domain. Pictorially, it means that the plot of \(X(e^{j\hat\omega})\) is “blurred” or “smoothed out” by \(\tilde{W}(e^{j\hat\omega}) e^{-j\hat\omega m}\).

  • In practice, we don’t usually calculate \(X(e^{j\hat\omega}; m)\) for every \(m\). Instead, we only need to at most step through the range of \(m\) of interest with a step size comparable to the lower bound on \(\sigma_t\) of \(\tilde{w}[n]\) because the lower bound gives the finest possible time resolution using the window \(\tilde{w}[n]\).

  • The table below gives some commonly used window signals. Note that the expression of each window signal \(w[n]\) given doesn’t centered at \(n=0\) (\(w[n]\) is the form usually given in textbooks). For the purpose of windowing in short-time DTFT, we use only odd-length windows. That is, we may obtain the \(0\)-centered window \(\tilde{w}[n]\) from the window \(w[n]\) provided by letting \(\tilde{w}[n] = w\left[ n + \frac{L-1}{2} \right]\), where \(L\) is the odd window length of \(w[n]\).

    Window name

    \(\boldsymbol{w[n]}~~\) for \(\boldsymbol{0 \leq n \leq L-1}\)

    Rectangular

    \(1\)

    Bartlett (triangular)

    \(\displaystyle 1 - \frac{2\left| n - \frac{L-1}{2} \right|}{L-1}\)

    Blackman

    \(\displaystyle 0.42 - 0.5\cos \left(\frac{2\pi n}{L-1}\right) + 0.08 \cos \left(\frac{4\pi n}{L-1}\right)\)

    Hamming

    \(\displaystyle 0.54 - 0.46 \cos \left(\frac{2\pi n}{L-1}\right)\)

    Hanning

    \(\displaystyle \frac{1}{2} \left[ 1 - \cos \left(\frac{2\pi n}{L-1}\right) \right]\)

    Kaiser

    \(\displaystyle \frac{I_0\left(\beta \sqrt{1-\left(\frac{2n}{L-1} - 1\right)^2}\right)}{I_0(\beta)}\)

    For the Kaiser window, \(I_0(\cdot)\) is the zeroth-order modified Bessel function of the first kind and \(\beta \geq 0\) is a parameter to control the shape of the window and the prominence of the sidelobes in the window’s DTFT. When \(\beta=0\), the Kaiser window reduces to the rectangular window.

  • The length \(L\) of the window signal determines the time resolution. A larger \(L\) gives poorer time but finer frequency resolution. The choice of the window signal is often based on a tradeoff between a smaller \(\sigma_{\omega}\) and the prominence of the sidelobes in the window’s DTFT for a chosen value of \(L\).