Frequency Domain and Fourier Transform

Frequency domain analysis is also called spectrum analysis or spectral analysis. Fourier transform is used to convert signals from time or space domain to the frequency domain. Signal information is converted to a magnitude and phase component of each frequency. The Fourier transform is converted to the magnitude of each frequency component squared, forming a power spectrum. This power spectrum is studied to determine which frequencies are present in the input signal and which are missing. The phase information can also be obtained from the Fourier transform. The cepstrum is a commonly used frequency domain transformation which converts a signal to the frequency domain through Fourier transform. It takes the Fourier transform (FT) of the logarithm of the estimated spectrum of a signal. The name cepstrum is devised by reversing the first four letters of the word spectrum. A very important property of the cepstral domain is that the convolution of two signals can be expressed as the addition of their cepstra.

x₁ * x₂ → x’₁ + x’₂

Signal Level of Transistor-Transistor/Low-power Schottky

Consider the signal-to-noise ratio results from the difference between the transmitting and the receiving level. Using the binary information representation, the signal-to-noise ratio can be adjusted to the environmental conditions within broad limits .

Signal level of a TTL-LS circuit
LOW level
Guaranteed transmitter level	max. 0.5 volt
Guaranteed receiver level	max. 0.8 volt
Static signal-to-noise ratio	0.3 volt
HIGH level
Guaranteed transmitter level	min. 2.7 volt
Guaranteed receiver level	min. 2.0 volt
Static signal-to-noise ratio	0.7 volt

 

Quantization Error Caused by A/D Conversion

Compared to an analog signal, a binary signal which represents only two states contains very little information. If a quantitiy to be represented digitally requires a wider range of values, it must be described by several bits.Analog quantities are processed digitally, when are converted into digital values first. An analog quantity can assume an infinite number of intermediate values while a digital quantity can only assume a limited number of values, therefore when analog signals are converted into discretized digital signals, quantization errors occur. Increasing the number of bits used for digital representation and the sampling rate of the analog signal reduces quantization errors. The complexity of data processing and transmission is increased with an increasing number of bits. The range of values must be adapted to the particular task, while choosing a binary representation that is not too extensive, in order to keep the loss of information during conversion as low as possible.

Quantization error caused by reduced discretization
and sampling rate

Determining the quantization error for displacement measurement:
Analog measuring range --- 0 to 30 cm
Range of values of an 8-bit unit --- 256
Quantization error --- (30/256) cm = 1.2 mm
Range of values of a 12-bit unit --- 4096
Quantization error --- (30/4096) cm = 0.073 mm

Digital Signal

Digital Signal is a signal which can only take discrete levels. A digitized analog signal or an arbitrary bit stream are digital signals. A signal generated by a modem, is in the first case considered as a digital signal, and in the second case as converted to an analog signal. A waveform that switches between two voltage levels representing the two states of a Boolean value (0 and 1) is referred to as a digital signal. The clock signal is a special digital signal that is used to synchronize digital circuits. The mathematical manipulation of an information signal to modify or improve it is called Digital signal processing. Digital signal transmission has many advantages over analog signal transmission. Applications of digital technology include the continuously growing number of PC’s, the communication network ISDN and the increasing use of digital control stations (Direct Digital Control: DDC).

Differential and Single-ended Signals

Two complementary paths carrying copies of the same signal in equal  and opposite polarity (relative to ground) are called Differential signals.

Differential Device Output

In Single-ended operation  there is only one path plus ground. It is a rather common architecture.

Single-Ended Device Output

Signal Formats

The most common signal formats are as follows:

• Non-Return-to-Zero (NRZ)
• Delayed Non-Return-to-Zero (DNRZ)
• Return-to-Zero (RZ)
• Return-to-One (R1)

For NRZ the waveform switches to a “1” and stays at that value until the next cycle boundary, when a valid bit occurs in the cycle. For DNRZ Similar to NRZ, the waveform switches to a “1” after a specified delay time, when a valid bit occurs in the cycle. For RZ the waveform switches to a “1,” then back to a “0” within the same cycle, when a valid bit occurs in the cycle. R1  assumes the cycle begins with a “1”, then switches to a “0” when the bit is valid, then switches back to a “1” before the cycle ends.