Analog frequency vs. Digital frequency

In the study of digital signal processing, many newly-started friends often confuse the analog frequency, the digital frequency and the relationship between them, even some friends who have some knowledge of digital signal processing are puzzled by this problem.

The frequency we normally refer to is the analog frequency, in hertz (Hz), or 1/s (1/s), in the case of no particular indication, and the mathematical notation is expressed in F. This is because the real-world signals are mostly analog signals, and the frequency is an important physical feature. The analog frequency expressed in Hertz represents the number of cycles per second in which the signal changes. If the unit circle is represented, 1, rotating a circle indicates that the signal changes one cycle, then the analog frequency refers to the number of cycles per second of the signal rotation.

Another concept in the analog frequency is the analog angular frequency, which is commonly represented by a mathematical symbol in radians per second (rad/s). From the point of view of the unit circle, the analog frequency is the number of rotations per second, and the angle change of each lap is 2pi. It is clear that the rotating F-ring corresponds to the radian of the 2pi*f. That

Ω=2pi*f (RAD/S) (1)

Digital signals are mostly sampled from analog signals, and sampling frequencies are typically expressed in FS. The more accurate term of the digital frequency should be normalized digital angular frequency, which is in radians (RAD), the mathematical symbol commonly used Ω. That

Ω=2pi*f/fs (RAD) (2)

The physical meaning is the number of radians that varies between adjacent two sample points, as shown in 1.

With the formula (1) and (2), we can switch freely between analog frequency and digital frequency. If there is a sine signal x[n], its frequency f=100hz, amplitude is a, the initial phase is 0, then the signal can be expressed as a formula:

X (t) =a*sin (2*PI*100*T)

Sampled by sampling frequency fs=500hz, the resulting digital signal x[n] is:

X[n] =a*sin (2*pi*100*n/fs) = A*sin (0.4*pi*n)

It is clear that the frequency of this digital signal is 0.4PI.

The above discussion shows that the analog frequency is not necessarily the same for two signals with exactly the same digital frequency as the sampling frequency is also considered here. This normalization has brought convenience, but also puzzled the understanding. In digital signal, although the sampling frequency does not appear explicitly, it is a bridge between analog signal and digital signal, which has a significant influence on the process of signal processing.

Analog frequency vs. Digital frequency