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Biomedical Signal and Image Processing
Just like in continuous signals, there is a significant amount of similarity among the
DFT and IDFT equations. This similarity causes the dual characteristics among
the signal in the discrete time and frequency domains. Other properties of the DFT
are very similar to the continuous FT except that in all cases integrals are replaced
by summations. Most of the properties of DFT are very similar to those of continuous FT and are explored in Problems section. Some other properties of DFT that are
unique to DFT are explored here.
2.4.1 PROPERTIES OF DFT
Some unique properties of DFT revolve around shift, repeat, and stretch of the
signal and the way these operations relate to convolution operation. These properties are explored in this section. First, we specify circular shift, repeat, and stretch
operations.
As shown in Figure 2.7, the circular shift transforms a discrete signal in
such a way that the time points in the far right-hand side are shifted to the
beginning of the signal in the left-hand side, and the rest of the time points are
simply shifted toward the right-hand side. The number of shift points is often
shown as the index of the operation, for example, a shift for two time points is
shown as SHIFT 2 .
The stretch operation, shown in Figure 2.8, allows stretching a signal by inserting zeros between consecutive time points in the signal. As in shift operation, the
number of stretch points is often shown as the index of the operation, for example,
a stretch for two time points is shown as STRETCH 2 . This operation is sometimes
referred to as zero-insertion operation.
Figure 2.9 represents another useful operation called repeat operation and represents repeating of a signal over an entire period. As in other operations, the number
of repeats is often shown as the index of the operation, for example, one complete
repeat is shown as REPEAT 2 .
g(n)
g(n – 1)
g(n – 2)
SHIFT 1 g(n)
SHIFT 2 g(n)
FIGURE 2.7 Circular shift of a signal g(n).
g(n)
STRETCH 2 g(n)
FIGURE 2.8 Stretch of a signal.
Biomedical Signal and Image Processing
Just like in continuous signals, there is a significant amount of similarity among the
DFT and IDFT equations. This similarity causes the dual characteristics among
the signal in the discrete time and frequency domains. Other properties of the DFT
are very similar to the continuous FT except that in all cases integrals are replaced
by summations. Most of the properties of DFT are very similar to those of continuous FT and are explored in Problems section. Some other properties of DFT that are
unique to DFT are explored here.
2.4.1 PROPERTIES OF DFT
Some unique properties of DFT revolve around shift, repeat, and stretch of the
signal and the way these operations relate to convolution operation. These properties are explored in this section. First, we specify circular shift, repeat, and stretch
operations.
As shown in Figure 2.7, the circular shift transforms a discrete signal in
such a way that the time points in the far right-hand side are shifted to the
beginning of the signal in the left-hand side, and the rest of the time points are
simply shifted toward the right-hand side. The number of shift points is often
shown as the index of the operation, for example, a shift for two time points is
shown as SHIFT 2 .
The stretch operation, shown in Figure 2.8, allows stretching a signal by inserting zeros between consecutive time points in the signal. As in shift operation, the
number of stretch points is often shown as the index of the operation, for example,
a stretch for two time points is shown as STRETCH 2 . This operation is sometimes
referred to as zero-insertion operation.
Figure 2.9 represents another useful operation called repeat operation and represents repeating of a signal over an entire period. As in other operations, the number
of repeats is often shown as the index of the operation, for example, one complete
repeat is shown as REPEAT 2 .
g(n)
g(n – 1)
g(n – 2)
SHIFT 1 g(n)
SHIFT 2 g(n)
FIGURE 2.7 Circular shift of a signal g(n).
g(n)
STRETCH 2 g(n)
FIGURE 2.8 Stretch of a signal.