Critically Sampled STFT Filter Bank

We will now analyze the filter bank interpretation of the STFT with hop size set to $R=M=N$ .

The kth subband signal in the DFT filter bank can be written as:

$$X_k(m) = \sum_{n=-\infty}^{\infty}h(mN - n)x(n)W_N^{kn}$$

with $W_N \equiv e^{-j2\pi/N}$ . The signal $X_k(m)$ is regarded as a complex sequence formed from the $k$ th DFT bin over time $m$ (in frames).

Making the change of variable $n=lN-r$ , the above equation becomes:

$$X_k(m) = \sum_{r=0}^{N-1}u_r(m)W_N^{kn} \protect$$ (11)

with

$$u_r(m) \equiv \sum_{l=-\infty}^{\infty}h(mN-lN+r)x(lN-r) \protect$$ (12)

Therefore, () can be interpreted as computing a length-$N$ FFT applied to the input block $[u_0(m) \cdots u_{N-1}(m)]^T$ .

Now, we'll form a simpler representation of the sequence $u_r(m)$ . First, define the polyphase decomposition of $h(n)$ and $x(m)$ :

$$p_i(m) = h(mN + i), \hspace{.5cm} i=0,1,...,N-1$$

If the filter $h(n)$ has length $M=LN$ , then each of its polyphase components $p_r(m)$ has length $L$ . Now define the sequence $x_r(m)$

$$x_r(m) \equiv x(mN - r), \hspace{.5cm} r=0,1,...,N-1$$

Finally, () can be expressed as:

$$u_r(m) = \sum_{l=-\infty}^{\infty}p_r(m-l)x_r(m)$$ (13)

Thus, every input bin to the DFT is actually a convolution between $x_r(m)$ and $p_i(m)$ , the $i$ th polyphase filter of the lowpass prototype filter, $h(n)$ .

• Complexity is now the cost of one full-rate convolution with $h(n)$ and the FFT cost.
• If $M \gg N$ ($L\gg1$ ), we can keep aliasing error within tolerable limits.
• If $M = 10N$ ($L=10$ ), it is possible to keep the reconstruction error below 0.1%.
• Similar development for the synthesis DFT bank

• Notice that when the polyphase filters are scalars (1-tap FIR filters) of unit gain, then this is simply a DFT block transform, with a rectangular window. The frequency response is a sinc, with poor frequency characteristics.
• By extending the length of the polyphase filters, $L>1$ , then the freqeuency characteristics of the window can become much better.
• The window length is no longer restricted to be of the same length as the transform.