Computing DFT by Hand and DFT Properties

DFT Properties (Circular, Not Linear!)

Property	Formula
Linearity	$a x_1[n] + b x_2[n] \leftrightarrow a X_1[k] + b X_2[k]$
Circular Shift	$x[(n-m)_N] \leftrightarrow W_N^{km} X[k]$
Circular Convolution	$x_1[n] \circledast x_2[n] \leftrightarrow X_1[k] X_2[k]$
Multiplication	$x_1[n] x_2[n] \leftrightarrow \frac{1}{N} X_1[k] \circledast X_2[k]$
Parseval	$\sum	x[n]	^2 = \frac{1}{N} \sum	X[k]	^2$

Critical: DFT convolution is circular, not linear! For linear convolution, zero-pad to length $\ge N_1 + N_2 - 1$.

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