Can the magnitude of a discrete-time Fourier transform be negative?

The answer is no, and let's take a look at why.

1. Can magnitude response be negative?
2. What is the magnitude of Fourier transform?
3. What does negative Fourier transform mean?

Can magnitude response be negative?

No. By definition, the magnitude of a complex number is always non-negative.

What is the magnitude of Fourier transform?

For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that frequency, and the argument of the complex value represents that complex sinusoid's phase offset. If a frequency is not present, the transform has a value of 0 for that frequency.

What does negative Fourier transform mean?

A negative real component just means the correlation against that particular cosine wave is negative, e.g. the input waveform seems to wiggle in the opposite direction of the corresponding cosine function, goes mostly low when the cosine goes high and vice versa. Same for the imaginary component and the sine function.