A fundamental problem in signal processing that we encountered in our study of radar systems is to estimate the frequency of a sinusoidal signal after observing the signal for a limited observation time. Suppose we have a system with the following parameters:
Expected range of observed sinusoidal frequencies:
0
≤
𝑓

Question

A fundamental problem in signal processing that we encountered in our study of radar systems is to estimate the frequency of a sinusoidal signal after observing the signal for a limited observation time. Suppose we have a system with the following parameters:
Expected range of observed sinusoidal frequencies: 
0
≤
𝑓
<
10
𝑘
𝐻
𝑧
Sampling rate: 
𝑓
𝑠
=
20
𝑘
𝐻
𝑧
⟶
 time between samples = 
50
𝜇
𝑠
𝑒
𝑐
.
Observation time: 
3.2
𝑚
𝑠
𝑒
𝑐
⟶
 
𝑁
=
64
 samples.
If the system shown below is used to estimate the frequency of the incoming sinusoid, how much zero padding is needed to guarantee that the error in the frequency estimate is no more than 
±
100
𝐻
𝑧
? Provide the minimum value of 
𝑀
 needed.

BlackTom AI · Accepted Answer

First, restate the setup to ensure we’re applying the right logic. We have a sinusoidal input sampled at f_s = 20 kHz, with an initial observation window of N = 64 samples. We can add zero padding of M zeros to obtain a total of N+M samples before performing the N+M point DFT to estimate the frequency.

Key idea 1: DFT frequency grid spacing. When you compute an N+M point DFT, the bin spacing in frequency is Δf = f_s / (N+M). This spacing determines how finely the spectrum is sampled in frequency domain.

Key idea 2: Frequency estimation accuracy from a discrete grid. If the true frequency lies between DFT bin centers, the best estimate you can achieve from the peak location is typically limited by the grid spacing. A common rule of thumb is that the maximum possible error in frequency estimation using a discrete grid is about ±Δf/2, since the true peak could lie halfway between two neighboring bins.

Therefore, to guarantee that the absolute estimation error is no more than Δf/2 ≤ 100 Hz, you need Δf ≤ 200 Hz.

Compute the minimum N+M that satisfies this bound. With f_s = 20,000 Hz,
Δf = f_s / (N+M) ≤ 200 → N+M ≥ f_s / 200 = 20,000 / 200 = 100.

We already have N = 64, so the required padding is M = (N+M) - N ≥ 100 - 64 = 36.

Hence, the minimum zero padding required is 36 samples to meet the ±100 Hz precision target.

Check intuition: Increasing M reduces Δf, giving a denser frequency grid. If M were smaller than 36, Δf would exceed 200 Hz, potentially yielding an estimation error larger than 100 Hz. Conversely, M = 36 yields N+M = 100 total points, giving Δf = 20,000 / 100 = 200 Hz and a worst-case error of about 100 Hz, which meets the requirement.

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