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You might be mixing two concepts, pertaining to the continuous or the discrete.

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in \mathbb{R}$ (for instance) or $e^{-j\omega n}$ with $\omega\in \mathbb{R}$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

You might be mixing two concepts, pertaining to (following Robert Bristow-Johnson)

• the "analog context" (more formally, the "continuous-time case"),
• the "digital context" (more formally the "discrete-time case").

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in \mathbb{R}$ (for instance) or $e^{-j\omega n}$ with $\omega\in \mathbb{R}$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

You might be mixing two concepts, pertaining to the continuous or the discrete.

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in [0,1]$ (for instance) or $e^{-j\omega n}$ with $\omega\in [0,2\pi]$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

You might be mixing two concepts, pertaining to the continuous or the discrete.

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in \mathbb{R}$ (for instance) or $e^{-j\omega n}$ with $\omega\in \mathbb{R}$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

You might be mixing two concepts, pertaining to the continuous or the discrete.

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in [0,1]$ (for instance) or $e^{-j\omega n}$ with $f\in [0,2\pi]$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

You might be mixing two concepts, pertaining to the continuous or the discrete.

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $e^{-j2\pi f n}$ with $f\in [0,1]$ (for instance) or $e^{-j\omega n}$ with $\omega\in [0,2\pi]$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.