# Why Fourier transform can be applied in digital signal processing?

Fourier transform can be used to analyze the frequency domain of a specific period of signal. However, I saw many of these transforms in textbooks are based on entire signal sequences, not processing them on a serial manner. For example, we have a sequence $$x[n]$$ and we can transform this sequence to $$X[k]$$. But many signal processing must be handling these signals sequentially, like processing Wi-Fi signals, LTE signals, etc.. Intuitively these signal must be processed when they received or with a little delay.

What is the real technique used in these contexts?

Regardless of block processing with the Fourier Transform, latency in any signal processing cannot be avoided given the laws of causality; any filtering done must have processing delay. For the Fourier Transform, this is minimized by processing the data in smaller blocks, which is ultimately trading time resolution and frequency resolution.

Since the OP mentioned Wifi, LTE and the Fourier Transform, and then questioned processing delay, I will focus on that in this answer. Wifi and LTE signals are indeed processed with the Fourier Transform, specifically the Discrete Fourier Transform using the FFT algorithm. The FFT is part of the modulation and demodulation using a format called Orthogonal Frequency Division Multiplexing (OFDM). In that, in the transmitter data is modulated onto sub-carriers, and these sub-carriers are grouped together as an FFT data block in the transmitter. This is converted to the time domain with an inverse FFT prior to transmission, and then in the receiver this data block is recovered and converted back to modulated sub-carrier with an FFT.

As suspected there is processing delay which is set by the FFT block size used and sampling rate. For example, in WIFI 802.11a the IFFT/FFT period is 3.2 us with a 0.8 us guard interval using a 64 point FFT with a sampling rate of 20 MSps. In LTE the FFT size can range from 128 bins to 2048 bins for channel bandwidths (and FFT sampling rates) from 1.25 MHz to 20 MHz. Overall latency for all processing is designed to be less than 100 ms, and this FFT block processing is part of that.

Fourier transform can be used to analyze the frequency domain of a specific period of signal. However, I saw many of these transforms in textbooks are based on entire signal sequences,

There are four closely related Fourier transforms.

1. An infinite length signal has a continuous frequency domain
2. A finite length signal has a discrete frequency domain
3. A continuous-time signal has an infinite frequency domain (this is a dual of 1, because the inverse Fourier transform is just a slightly modified Fourier transform)
4. A discrete-time signal has a finite frequency domain (again, this is a dual of 2, for the same reason).
time length frequency length name
continuous infinite continuous infinite Fourier Transform*
continuous finite discrete infinite Fourier Series
discrete infinite continuous finite Discrete-time Fourier transform
discrete finite discrete finite Discrete Fourier Transform**

* "Plain old" Fourier transform

** AKA "DFT", it's what can be made fast for the FFT

Your confusion probably arises because the DFT/FFT is heavily used in signal processing algorithms, which are realized in hardware. At the same time, the first three transforms in the table above are used strictly for analysis -- analysis, in turn, is used heavily in designing digital signal processing algorithms.

So you'll see "the Fourier transform" mentioned, but there's four flavors. In signal processing, the one that's actually used in algorithms is the FFT, because it's the only one that makes sense with sampled, finite-length data.

• Today I learned that Stackexchange previews will show tables, but Stackexchange answers won't. How -- convenient. Dec 13, 2022 at 16:00
• That seems to have fixed it. I removed the back quote end pieces around the table.
– Peter K.
Dec 13, 2022 at 16:21
• That's weird -- I put them there after it didn't render for me. Dec 13, 2022 at 17:20
• But -- problem solved. Dec 13, 2022 at 17:20