Sampling theory
A continuous time random signal is shown. Based on this a number of important concepts are shown below. The signal is continuous time signal with continuous amplitude. This type of signal is known as an analog signal.
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→Time
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nT→
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1T
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→Time
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x(n)→
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5.5
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8.4
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7.3
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←Sampled signal. Discrete-time
signal - time is discrete, amplitude is continuous.
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←Quantized. Quantization noise
(error). Digital signal - both time and amplitude are discrete.
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101
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010
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011
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001
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111
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111
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Encoded to 3 bits/sample.
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↑Note this particular point exhibits saturation (out of
range). Rounded down to 7, not 8.
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If we were to represent sample value with infinite precision then we would require registers and memory words of arbitrarily large size. But, because of a finite word length we round off the sample values. This introduces the quantization noise.
The process of generating a discrete-time signal from the analog signal is shown in block diagram. In the digital signal processing course we are mostly dealing with discrete-time rather than digital signals and systems, the latter is a subset of the former.
The 3 boxes shown in the above diagram can be represented by an analog to digital converter (ADC). A complete digital signal processing (DSP) system comprises of an ADC, a DSP algorithm
(for example a difference equation) and a digital to analog converter (DAC) which is shown below.
As the name implies discrete-time signals are defined only at discrete instants of time. Discrete-time signals can arise by sampling analog signals such as a voice signal or a temperature signal in telemetry. Discrete-time signals may also arise naturally, e.g., the number of cars sold on a specific day in a year, or the closing DJIA figure for a specific day of the year.
AT&T's T1 Stream The voice signal is band limited to 3.3 kHz, sampled at 8000 Hz (8000 samples per second), quantized and encoded into 8 bits per sample. Twenty four such voice channels are combined to form the T1 stream or signal.
Sampling interval = 1/8000 Hz = 0.125 msec.
Bit rate for each channel = 8000 samples/sec x 8 bits/sample = 64000 bits/sec.
Bit rate for T1 = 64000 bits/sec per channel x 24 channels = 1 544 000 bits/sec.
Commercial examples CD, Super Audio CD (SACD), DVD Audio (DVD-A), Digital audio broadcasting - 32 kHz, and Digital audio tape (DAT) - 48 kHz.
Commercial Audio Examples
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CD
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Super Audio CD
(SACD)
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DVD-Audio
(DVD-A)
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Sampling
Rate
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44.1 kHz
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2.8224 MHz
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44.1, 88.2 or
48, 96, 192 kHz
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Coding
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16-bit PCM per sample
With 2 channels the bit rate is 1.4112
Mbits/sec, but additional error control bits etc., raise it to 4.3218 Mbits/sec.
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1-bit DSD (Direct Stream Digital - like Delta modulation)
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12-, 20-, 24- bit
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