Let $x[n]$ be a infinite-length discrete-time signal with $x[n] = 0$ for $0 \leq n \leq N$ . We can compute the discrete DFT $X[k]$ and the continuous DTFT $X(\omega)$. The DFT is a sampled version of the DTFT, i.e., $$ X[k] = X(2\pi k / N). $$ Please also see posts here and here. Now, we consider the maximum magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.

*What is the relation between these maximum values? Is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$? *

Let $x[n]$ be a finite-length discrete-time signal with length $N$. The continuous DTFT $X(\omega)$ is then $$ X(\omega) = \sum_{n = 0}^{N-1} x[n] e^{-j \omega n}. $$

The length-$N$ DFT of $x[n]$ is $$ X[k] = \sum_{n = 0}^{N-1} x[n] e^{-j 2 \pi \frac{k n}{N}}. $$ For this, the DFT is a sampled version of the DTFT, i.e., $$ X[k] = X(2\pi k / N). $$ Please also see posts here and here. Now, we consider the maximum magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.

What is the relation between these maximum values? Is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$?

Let $x[n]$ be a finite-length discrete-time signal of length $N$. We can compute the discrete DFT $X[k]$ and the continuous DTFT $X(\omega)$. The DFT is a sampled version of the DTFT, i.e., $$ X[k] = X(2\pi k / N). $$ Please also see posts here and here. Now, we consider the maximum peak magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.

What is the relation between these peak values?

For example, is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$? I'm also interested not only in the largest possible difference, but a statistically expected difference, for instance when $x[n]$ is from a Gaussian distribution.

Let $x[n]$ be a infinite-length discrete-time signal with $x[n] = 0$ for $0 \leq n \leq N$ . We can compute the discrete DFT $X[k]$ and the continuous DTFT $X(\omega)$. The DFT is a sampled version of the DTFT, i.e., $$ X[k] = X(2\pi k / N). $$ Please also see posts here and here. Now, we consider the maximum magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.

*What is the relation between these maximum values? Is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$? *