Discrete frequency

Discrete Frequency is defined as the frequency with which the samples of a discrete sinusoid occur. Just as in its continuous-time counterpart (see frequency), the discrete time signal has a time axis, conventionally denoted by "n". The time variable "n", however, has a constraint that its continuous time clone does not. The variable "n" can take on only discrete integral values, and thus is not a continuous variable.

Discrete Time Sinusoids

Discrete time sinusoids are characterized by the mathematical relationship between amplitude and the discrete number "n", as :

$x=Acos\; (omega\; n)\; mbox\{,\; where\; \}\; omega=2pi\; f\; mbox\{\; and\; \}\; n\; in\; mathcal\{Z\}$

Here, "f" is termed the discrete frequency of the sinusoid. "w" is the angular frequency. The frequency "f" has the dimensions of cycles per sample. This fact can relate to its continuous time counterpart by considering "sample" as the unit of the 'time' axis "n", instead of seconds in time axis "t", as is done in continuous time.

One can use the following MATLAB m-code to generate discrete sinusoids with different fundamental periods:

%This m code plots a discrete time sinusoid with the specified fundamental%period N and integral value k, to give f=k/Nn=0:1:100;N=25;k=1;y=cos(2*pi*n*k/N);stem(y);

Properties of Discrete Time Sinusoids

Discrete time sinusoids have the following properties:

Periodicity

$x(n+mathcal\{N\})=x(n)\; mbox\{\; for\; all\; \}n$The smallest value of "N" for which this equation is true is called the fundamental period of the sinusoid. Proof of this can be obtained by simple trigonometric manipulations:

$cos(2\; pi\; f\; (mathcal\{N\}+n))=cos(2\; pi\; f\; n)$

for this to be true, there must exist an integer "k" such that,

$2\; pi\; f\; mathcal\{N\}=2\; k\; pi$

Thus, a discrete time sinusoid is periodic only if its frequency can be expressed as a ratio of two integers:

$f=frac\{k\}\{mathcal\{N$

Aliased Frequencies

Discrete time sinusoids whose frequencies are separated by an integer multiple of $2pi$ are identical. This follows as under:

$cos((omega\_\{0\}+2pi)n)=cos(omega\_\{0\}\; n)mbox\{\; \}\; forall\; ninmathcal\{Z\}$

Thus, all sinusoidal frequencies $omega\_\{k\}$ are indistinguishable. Where,

$omega\_\{k\}=omega\_\{0\}+2\; k\; pimbox\{,\; where\; \}\; -pile\; omega\_\{0\}le\; pi$

Any sinusoid with an angular frequency that falls outside the interval $-pimbox\{\; to\; \}pi$ is identical to a sinusoidal frequency that falls within the fundamental interval. This is called aliasing of frequency; any frequency outside the above period is an alias of some frequency inside the period. Thus $-pileomegalepi$ is regarded as the period of unique frequencies, and is said to contain "all the discrete frequencies" in contrast to the continuous time frequencies which range from $-inftymbox\{\; to\; \}infty$. As a consequence, the highest rate of oscillation in a discrete time sinusoid occurs when $omega=pmpimbox\{\; or\; \}f=pmfrac\{1\}\{2\}$.

ampling & Discrete Frequencies

The apparent problem in this regards is the fact that the highest oscillation in discrete time occurs when frequency is 0.5 or -0.5. How then can discrete time systems be used (and are being used) to process even the lowest frequencies like audio, let alone work with satellite communication receivers at frequencies in the GHz (1,000,000,000 Hz) range? The answer to the question lies in visualizing the concept of discrete time as a physical phenomenon and not just a mathematical constraint. The 'time' operator "n" is not to be visualized as a discrete number that can take on values in seconds, but as a number that can take discrete values in any time unit: seconds, milliseconds, microseconds, nanoseconds, picoseconds etc. Thus it can be understood how discrete waveforms can attain higher frequency.

ampling of a continuous-time signal

As seen in the block diagram, the sampler 'allows' the analog input signal to pass at only certain discrete intervals. This interval, the sampling period, defines the final frequency of the samples at the output. Thus, no matter how high the input analog frequency is, if the time duration between two subsequent samples is made sufficiently small, then these high frequencies can also have their discrete time counterparts.

Frequency ranges

Say, for a continuous time signal "x(t)", sampling is done at a rate Fs(=1/Ts).Thus the 'samples' at the output side are:

$x(nT)=x(n)=Acos(2\; pi\; n\; F\; T)=Acos(frac\{2pi\; n\; F\}\{F\_s\})$

where,$x(t)=Acos(2\; pi\; F\; t)$

As it can be seen, the relation of the continuous frequency to the discrete frequency can be established as:

$f=frac\{F\}\{F\_s\}mbox\{\; or\; \}\; omega=Omega\; T$

Now, the range for discrete frequency is:

$frac\{-1\}\{2\}le\; f\; lefrac\{1\}\{2\}$

Thus,

$frac\{-1\}\{2T\}=frac\{-F\_s\}\{2\}le\; F\; lefrac\{F\_s\}\{2\}=frac\{1\}\{2T\}mbox\{\; where\; F\; is\; analog\; frequency\}$

This is the famous constraint given by the sampling theorem (nyquist theorem), that the sampling frequency has to be at the least double of the highest frequency of the input signal. In conclusion, one can say that the discrete frequencies themselves have values in the range [-0.5,0.5] only, but the rate at which the samples occur (sampling rate), or the rate at which a continuous time signal can be 'converted' into a discrete time signal, depends on the sampling frequency. In practice, majority of the discrete signals encountered are either sampled from some analog input, or are destined to be converted into analog signals at the end of processing.

References

"Digital Signal Processing: Principles, Algorithms and Applications", "by" John G. Proakis & Dimitris G. Manolakis. 4th ed., Prentice-Hall.

See also

*Digital signal processing
*nyquist theorem
*Aliasing
*Fourier Transform
*Discrete-time Fourier transform
*Discrete Fourier Transform
*FFT