# Timeshifting in streaming based audio processing

I am currently running into a larger problem whilst programming an audio processing software. My target is time-shifting signals in frequency domain, then transform them back to time domain and merging 2 streams into an output.

I'm windowing the incoming signal(s) to extract frames of 32 samples with an offset of 16 samples. Hence, assuming 64 input (and output) samples, a total of 4 overlapping frames is created:
Frame 1 =$$[-16...-1,0...15]$$,
Frame 2 = $$[0...31]$$,
Frame 3 = $$[16...47]$$,
Frame 4 = $$[32...63]$$.
Here $$[-16...-1]$$ are samples from a previous iteration (or zeroed out in the beginning) and the last 16 samples $$[48...63]$$ are stored for the next iteration.

Assume $$x[n]$$ is the input signal in the first frame, and it is timeshifted and added to output such that $$x[n+2]=y[m]$$. For that im adding leading and trailing zeros before shifting. However, it would shift the signal out of the scope of the output buffer. Same for frame 4 and timeshifts like $$[n-2]$$, where it shifts out of the buffer (into the next iteration).

How can I handle such shifting, considering i have fixed shift values, e.g. 6 samples?

Any hint appreciated.
The picture shows how unshifted frames are assembled in the output.

• why would you do this in frequency domain? An integer sample shift is just exactly that in time domain! Sep 5 '21 at 16:00
• Other processing takes place in frequency domain. It's however somewhat irrelevant to the problem description, because in the end it means processing audio samples either way. Sep 5 '21 at 16:06
• so if you're multiplying the frequency-domain data with $e^{-j 2 \pi f \tau}$ to effect a delay of $\tau$, you gotta be careful with this. it's because everything you do with the DFT is circular and your timeshifting will be circular and there can be time-aliasing, if you don't do something about it. Sep 6 '21 at 1:03
• @robertbristow-johnson , i suppose "do something about it" means zeropadding? Sep 6 '21 at 13:08
• yeah, probably. either that or recognize the aliasing at the edges. but zero-padding is the only way i know of to completely avoid time aliasing that the FFT will otherwise put in there. Sep 6 '21 at 13:19

## 1 Answer

My target is time-shifting signals in frequency domain

That's rather complicated. It's MUCH easier to time shift directly in the time domain.

If you absolutely must do this in the frequency domain, you can time shift the a buffer by multiplying with $$H(k) = e^{-j2\pi\frac{kn}{N}}$$ where j is frequency, n amount of delay in samples and N the FFT length.

However that shift is circular, not linear. In order to get a linear shift you will need to increase your buffer size, zero pad to ensure there is no-wrap around, and manage the overlaps between buffers properly.

• I suppose j is the imaginary unit and k is frequency ;) However I already have shifted the signals and can see the shift in my buffers and plot them quite nicely using matlab. But I'm still left with signals with more than 64 samples (output buffer size), and figuring out the overlaps between the buffers and merging them to the output is exactly where the trouble begins. Sep 6 '21 at 13:05
• j is the imaginary unit and k is the frequency index. Sorry , I don't understand where your problem is. Can you add some math with what exactly are you doing and where does it not work ? Sep 7 '21 at 16:41